QB 

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UC-NRLF 


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AZIMUTH 


BY 


GEORGE    L.    HOSMER 

ASSOCIATE   PROFESSOR   OF  TOPOGRAPHICAL  ENGINEERING 
MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY 


SECOND  EDITION 
REVISED 


NEW   YORK 

JOHN   WILEY  &   SONS,    INC. 

LONDON:   CHAPMAN  &  HALL,  LIMITED 

1916 


ft* 

\-1-u 


Copyright,  1909  and  1916, 

BY 
GEORGE   L.  HOSMER 


TYI>OGRAPHY    AND    ELBCTROTYPING   BY 

F.  H.  GILSON  COMPANY 
BOSTON,  MASS. 

PRHSSWORK  BY 

BERWICK    &   SMITH   CO. 
NORWOOD,   MASS. 


PREFACE. 


THE  purpose  of  this  volume  is  to  present  in  compact  form  certain 
approximate  methods  of  determining  the  true  bearing  of  a  line,  together 
with  the  necessary  rules  and  tables  arranged  in  a  simple  manner  so  that 
they  will  be  useful  to  the  practical  surveyor.  It  is  a  handbook  rather 
than  a  text-book,  hence  many  subjects  have  been  wholly  omitted  which 
are  ordinarily  included  in  books  on  Practical  Astronomy  but  which  are 
not  essential  in  learning  to  make  the  observations  described  in  this  book. 
In  all  of  the  methods  here  treated  the  object  sought  is  to  secure  sufficient 
accuracy  for  the  purpose  of  checking  the  measured  angles  of  a  survey 
with  the  least  expenditure  of  time.  For  this  reason  many  approxima- 
tions have  been  made  and  many  refinements  omitted  which  simplify  the 
calculations  without  introducing  serious  error  into  the  results,  and 
although  such  a  treatment  would  scarcely  be  proper  in  a  text-book  the 
gain  in  simplicity  and  convenience  would  seem  to  justify  its  use  in  a  book 
of  this  character. 

The  necessity  for  making  astronomical  observations  for  azimuth  is 
confined  chiefly  to  geodetic  work,  and  arises  so  seldom  in  general  engi- 
neering practice  that  many  persons  engaged  in  surveying  are  not  familiar 
with  astronomical  methods  and  will  not  feel  confident  of  obtaining  reliable 
results,  and  therefore  are  likely  to  avoid  making  use  of  such  observations 
even  when  they  might  be  of  great  practical  value.  This  consideration 
together  with  the  fact  that  the  rigorous  methods  of  calculating  azimuth 
are  rather  long  and  complex,  have  tended  to  prevent  astronomical  obser- 
vations from  being  more  generally  applied  to  surveying.  The  author 
has  endeavored  to  so  present  the  subject  that  a  person  who  is  unfamiliar 
with  astronomy  will  be  able  to  apply  these  methods  and  obtain  satisfactory 
results  without  taking  the  time  to  completely  master  the  theory  under- 
lying the  method  used.  The  rules  and  tables  have  been  put  in  compact 
form  so  that  the  book  may  be  carried  in  the  field  and  the  results  of  obser- 
vations worked  up  at  once  if  desired.  The  value  of  having,  from  time 
to  time,  an  independent  check  on  the  angular  measurements  of  an  exten- 
sive survey  will  certainly  warrant  spending  a  few  minutes'  time  in  making 
observations  and  computing  the  results. 

The  methods  here  presented  are  not  new  but  have  all  appeared  in  one 
form  or  another  in  works  on  Navigation,  Astronomy,  and  Surveying. 


359274 


iv  PREFACE 

Much  valuable  matter  written  on  this  subject,  however,  is  so  scattered 
that  it  is  difficult  to  find  in  one  small  book  all  that  would  be  needed  by 
the  surveyor  in  making  azimuth  observations. 

The  author  desires  to  acknowledge  his  indebtedness  to  Professor 
C.  F.  Allen  for  the  use  of  the  electrotype  of  Table  XVIII  taken  from  his 
"  Field  and  Office  Tables,"  and  to  Professor  F.  E.  Turneaure  for  permis- 
sion to  reprint  Table  V  from  Johnson's  "  Theory  and  Practice  of  Survey- 
ing." Thanks  are  due  to  Professor  C.  B.  Breed  for  valuable  suggestions 
and  criticism  of  the  manuscript. 

G.  L.  H. 

BOSTON,  MASS.,  January,  1909. 


PREFACE   TO    THE   SECOND    EDITION. 


IN  the  present  edition  the  following  changes  have  been  made :  (i)  A 
new  method  is  given  for  finding  the  azimuth  by  an  ^observation  on  the 
pole  star  at  any  hour  angle  when  the  local  time  is  known.  This  will  be 
found  convenient  by  those  who  prefer  observations  on  Polaris  to  direct 
solar  observations.  (2)  The  tables  of  the  sun's  declination  have  been 
extended  to  1919  and  rules  have  been  added  for  computing  the  declina- 
tions for  subsequent  years.  (3)  The  star  maps  are  new  and  are  more 
complete  than  those  of  the  first  edition. 

G.  L.  H. 

June,  1916. 


CONTENTS. 


ART.  PAGE 

1.  CHECKING  THE  ANGLES  OF  A  SURVEY i 

2.  SUN  AND  STAR  OBSERVATIONS i 

3.  APPARENT  MOTIONS  OF  THE  STARS  —  MERIDIAN  . 2 

4.  POLAR  DISTANCE  —  DECLINATION 2 

5.  HOUR  ANGLE 3 

6.  LATITUDE  AND  ELEVATION  OF  POLE 3 

7.  CORRECTING  AN  ALTITUDE  —  REFRACTION  CORRECTION  ....  3 

8.  INDEX  CORRECTION 3 

9.  MAKING  SOLAR  OBSERVATIONS 4 

10.  MAKING  STAR  OBSERVATIONS 4 

11.  AZIMUTH  MARK 5 

12.  CONVERGENCE  OF  MERIDIANS 5 

METHODS  OF  OBSERVING 

13.  AZIMUTH  BY  AN  OBSERVED  ALTITUDE  OF  THE  SUN 6 

14.  COMPUTING  THE  AZIMUTH 8 

15.  WHEN  TO  OBSERVE 10 

16.  AZIMUTH  BY  AN  ALTITUDE  OF  A  STAR 13 

17.  OBSERVATIONS  FOR  LATITUDE 13 

18.  LATITUDE  BY  THE  SUN  AT  NOON 13 

19.  AZIMUTH  AND  LATITUDE  BY  OBSERVATION  ON  3  CASSIOPEIA 

AND  POLARIS 15 

20.  FINDING  THE  STARS 15 

21.  EXPLANATION  OF  THE  METHOD 15 

22.  THE  TABLES 19 

23.  MAKING  THE  OBSERVATIONS 20 

24.  OBSERVATIONS  ON  d  DRACONIS 23 

25.  MERIDIAN  LINE  BY  POLARIS  AT  CULMINATION 25 

26.  AZIMUTH  BY  POLARIS  WHEN  THE  TIME  is  KNOWN 26 

27.  ACCURATE  DETERMINATION  OF  AZIMUTH  BY  POLARIS 27 

28.  DETERMINING  THE  HOUR  ANGLE 28 

29.  DETERMINING  THE  AZIMUTH 29 

30.  COMPUTING  THE  AZIMUTH 29 

31.  MERIDIAN  BY  POLARIS  AT  ELONGATION 33 

32.  MERIDIAN  BY  EQUAL  ALTITUDES  OF  A  STAR 35 

33.  MERIDIAN  BY  EQUAL  ALTITUDES  OF  THE  SUN 37 

TABLES 39-73 

v 


AZIMUTH. 


1.  Checking  the  Angles  of  a  Survey  by  Astronomical  Azimuths.  —  In 

the  following  pages  are  given  several  short  and  convenient  methods  of 
determining  the  azimuth  of  a  line  with  an  engineer's  transit.  While 
these  methods  may  be  used  to  determine  an  azimuth  for  any  purpose 
which  does  not  require  great  precision,  the  formulae  and  the  tables  have 
been  specially  arranged  so  that  it  will  be  practicable  to  compute  the 
azimuth  in  the  field  for  the  purpose  of  checking  the  angles  of  a  survey. 
In  a  preliminary  railroad  survey,  for  instance,  or  in  running  long  traverses 
by  the  stadia  method,  there  will  ordinarily  be  no  reliable  check  on  the 
measured  angles  such  as  that  obtained  by  closing  a  circuit  or  by  connect- 
ing the  survey  with  some  line  of  known  azimuth.  In  such  cases  the 
angles  can  be  checked,  with  all  the  accuracy  required,  by  determining  the 
azimuth  of  some  line  of  the  survey  either  by  means  of  a  sun  observation 
or  by  an  observation  on  the  pole-star,  and  comparing  the  azimuth  thus 
determined  with  the  azimuth  computed  by  means  of  the  measured  angles. 
With  convenient  tables  the  azimuth  may  be  computed  in  the  field  in  a 
few  minutes'  time,  so  that  it  will  be  known  at  once  whether  the  preceding 
angles  are  correct. 

The  methods  explained  in  Arts.  13  to  24  inclusive  will  give  results 
sufficiently  accurate  for  a  check  of  the  angles  of  an  ordinary  survey,  but 
in  these  methods  extreme  accuracy  is  sacrificed  to  convenience  and 
rapidity  in  the  computations  in  order  that  results  may  be  quickly 
obtained  in  the  field.  In  Art.  27  is  given  a  more  accurate  method  which 
may  be  used  when  it  is  necessary  to  obtain  an  azimuth  that  is  correct 
within  a  few  seconds. 

2.  Sun  and  Star  Observations.  —  With  regard  to  the  relative  advantages 
of  sun  observations  and  observations  on  the  pole-star  it  may  be  said  that 
observations  on  the  sun  are  the  more  convenient  of  the  two,  but  will  not 
give  results  of  great  accuracy;  their  particular  advantage  is  that  they  can 
be  made  while  the  survey  is  in  progress  and  with  the  loss  of  but  a  few 
minutes'  time,  and  if  the  azimuth  is  desired  only  to  about  one  minute  of 
angle,  sun  observations  will  be  sufficiently  accurate.     By  observations  on 
the  pole-star  the  azimuth  may  be  determined  with  great  accuracy,  and 
such  observations  are  the  ones  most  commonly  employed  when  the  best 

i 


2  AZIMUTH 

results  ?re  oougrr..  Sta*-  ob^rvuiic/ns,  however,  have  the  disadvantage 
that  they  must  be  made  at  night,  in  which  case  the  surveyor  has  to  make 
a  special  trip  to  the  point  of  observation,  and  must  carry  on  his  work 
under  various  practical  difficulties  which  are  not  encountered  in  making 
sun  observations.  If  a  precise  azimuth  is  desired  it  is  necessary  to  observe 
on  the  pole-star,  or  some  other  star  close  to  the  pole,  and  also  to  make 
auxiliary  observations  for  latitude  and  hour  angle,  on  which  the  com- 
puted azimuth  depends.  The  data  for  such  an  observation  must  be 
obtained  from  the  Nautical  Almanac.  If,  however,  only  an  approximate 
result  is  desired,  say  within  about  i  minute,  the  observations  may  be 
made  on  the  pole-star  by  the  method  given  in  Arts.  19  to  24,  and  the 
azimuth  found  by  Tables  VII  to  XII.  This  method  has  the  advantages 
that  no  Nautical  Almanac  is  required  and  that  there  is  little  calculation 
except  interpolation  in  the  tables. 

3.  Apparent  Motions  of  the  Stars  —  Meridian.  —  If  one  watches  the 
stars  for  several  hours  he  will  observe  that  they  all  appear  to  move  from 
east  to  west  in  circular  paths  as  though  they  were  all  attached  to  the 
surface  of  a  great  sphere,  and  this  sphere  were  turning  about  an  axis, 
the  earth  being  at  the  centre  of  the  sphere.     Those  stars  which  are  near 
the  equator  all  move  in  large  circles.     As  the  observer  looks  farther  north 
he  sees  that  these  circles  grow  smaller,  their  common  centre  being  an 
imaginary  point  called  the  pole.     A  vertical  plane  through  the  pole  cuts 
out  on  the  sphere  a  great  circle  called  the  meridian  of  the  observer.     The 
line  in  which  this  meridian  plane  cuts  a  horizontal  plane  through  the 
observer  is  called  the  meridian  line.     At  a  distance  of  about  i°  10'  from 
the  north  pole  is  a  bright  star  which  moves  around  the  pole  in  a  very  small 
circle  and  is  known  as  the  pole-star,  or  Polaris.     Since  its  circle  is  small 
its  apparent  motion  is  very  slow,  and  consequently  it  is  easy  to  determine 
with  accuracy  its  true  bearing  at  any  instant  and  to  measure  an  angle 
from  the  star  to  a  reference  mark  on  the  ground. 

The  relative  position  of  the  fixed  stars  *  is  always  practically  the  same, 
and  the  pole  stays  nearly  in  the  same  position  among  the  stars  year  'after 
year.  We  may  think  of  all  of  the  stars  in  the  north  as  moving  in  circles 
around  the  pole  once  each  day,  the  size  of  the  circle  of  any  star  and  the 
speed  of  the  star's  motion  depending  upon  how  far  that  star  is  from  the 
pole. 

4.  Polar  Distance  —  Declination.  —  The  angular  distance  to  a  star 
from  the  pole  is  known  as  its  polar  distance,  an  angle  which  changes 
slightly  from  year  to  year  and  may  be  obtained  for  any  date  from  the 

*  The  term  fixed  star  is  used  to  distinguish  the  very  distant  stars  from  the  planets;  the 
latter  are  within  the  solar  system  and  consequently  appear  to  change  their  positions 
rapidly.  The  fixed  stars  have  but  a  slight  motion,  imperceptible  to  the  naked  eye. 


INDEX  CORRECTION  3 

American  Rphemeris  and  Nautical  Almanac.*  In  case  of  the  sun  or  a  star 
which  is  far  from  the  pole  it  is  more  convenient  to  define  its  position  by  its 
angular  distance  from  the  equator,  i.e.,  by  the  declination,  which  is  the 
complement  of  the  polar  distance. 

5.  Hour  Angle.  —  The  hour  angle  of  a  body  is  the  number  of  hours, 
minutes,  and  seconds  that  have  elapsed  since  the  body  was  on  the  obser- 
ver's meridian.     Hence  it  is  simply  the  angle  through  which  the  body 
has  appeared  to  move  since  it  passed  the  meridian  of  the  observer.     It 
will  be  seen  then  that  a  star  which  has  an  hour  angle  between  oh  and  12^ 
is  west  of  the  meridian;  if  the  hour  angle  is  between  12*1  and  24^  the  star 
is  east  of  the  meridian.     Hours  or  degrees  are  simply  units  of  measure- 
ment of  a  circumference;  in  one  case  the  circle  is  divided  into  24  hours, 
and  in  the  other  case  into  360  degrees.     Hence  an  hour  angle  which  is 
expressed  in  hours,  minutes,  and  seconds  of  time  may  be  converted  into 
degrees,  minutes,  and  seconds  of  arc.     Since  24^  =  360°  we  have  ih  =  15°, 
or,  in  a  more  convenient  form,  i°  =  4m  and  i'  =  48. 

6.  Latitude  and  Elevation  of  Pole.  —  A  very  important  principle  in 
astronomy  is  that  the  angular  altitude  of  the  pole  above  an  observer's 
horizon  equals  the  latitude  of  the  observer.     Hence  the  latitude  may  be 
determined  by   simply   finding,    by    some    means,   the    altitude   of    the 
pole  above  the  horizon.    It  may  also  be  determined  by  finding  the  merid- 
ian altitude  of  a  point  on  the  equator,  which  is  the  complement  of   the 
latitude. 

7.  Correcting  an  Altitude — Refraction  Correction.  —  In   measuring 
an  altitude  of  any  heavenly  body  it  is  necessary  to  apply  a  correction  to 
the  measured  altitude  to  allow  for  the  bending  (refraction)  of  the  rays  of 
light   in  passing   through  the  earth's   atmosphere.     This   correction   is 
always  subtracted  from  the  observed  altitude  to  reduce  it  to  the  true 
altitude,  because  the  ray  of  light  is  concave  downward,  and  hence  the 
object  appears  too  high  above  the  horizon.     This  correction  in  minutes 
of  angle  may  be  taken  from  Table  I. 

8.  Index  Correction.  —  In  measuring  altitudes  with  a  transit  the  plate 
bubbles  should  not  be  relied  upon,  but  after  an  altitude  has  been  measured 
the  vertical  arc  should  be  examined  to  see  if  the  vernier  reads  o°  when 
the  telescope  bubble  is  in  the  middle  of  its  tube.     If  it  does  not  read  o°  the 
reading  of  the  vertical  arc  must  be  corrected  by  an  amount  equal  to  this 
error.     If  the  o  line  of  the  vernier  is  on  the  same  side  of  the  o°  line  of  the 
arc  at  both  readings,  the  index  correction  is  to  be  subtracted;  if  the  o  line 
of  the  vernier  passes  the  o°  graduation  when  the  telescope  is  brought 
down  to  the  horizontal  position  the  correction  is  to  be  added. 

*  Published  annually  (three  years  in  advance)  by  the  Bureau  of  Equipment,  Navy 
Department,  Washington,  D.  C. 


4  AZIMUTH 

9.  Making  Solar  Observations.  —  In  making  observations  on  the  sun 
with  the  surveyor's  transit  it  is  desirable  to  have  a  dark  glass  placed  over 
the  eyepiece  to  cut  down  the  light  so  that  it  will  not  be  too  bright  for  the 
eye.     If  no  such  dark  glass  accompanies  the  instrument  the  observation 
can  be  made  by  throwing  the  sun's  image  on  a  piece  of  paper  held  behind 
the  eyepiece.     If  the  objective  is  focussed  on  a  distant  object  and  the 
eyepiece  tube  drawn  out,  the  image  of  the  sun  and  the  cross-hairs  can  be 
seen  on  the  paper.     The  disc  can  be  sharply  focussed  by  moving  the 
paper  toward  or  away  from  the  eyepiece.     By  this  device  observations 
can  be  made  almost  as  well  as  by  means  of  the  dark  glass. 

10.  Making  Star  Observations.  —  In  making  observations  at  night  it  is 
necessary  to  illuminate  the  field  of  view  of  the  telescope  in  order  to  make 
the  cross-hairs  visible.     This  should  be  done  in  such  a  manner  as  to 
avoid  heating  either  the  instrument  or  the  air  just  in  front  of  the  telescope. 
If  the  instrument  is  heated  the  adjustments  will  be  disturbed;  if  the  air  is 
heated  the  image  will  appear  very  unsteady.     With  some  instruments 
there  is  a  special  reflector  placed  in  a  shade  tube,  fitting  to  the  objective 
slide,  so  that  when  a  lantern  is  held  at  one  side  of  the  telescope  the  light 
is  reflected  into  the  tube.     If  no  such  reflector  is  at  hand  a  satisfactory 
result  may  be  obtained  by  placing  a  piece  of  tracing  cloth,  or  oiled  paper, 
in  front  of  the  objective,  folding  the  edges  back  over  the  tube  and  fastening 
it  in  place  by  means  of  a  rubber  band.     A  hole  about  one-half  inch  in 
diameter  should  be  cut  in  the  centre  of  the  paper  so  that  light  from  the 
star  can  pass  through  the  central  portion  of  the  lens  while  the  outer  edge 
of  the  lens  is  covered.     The  cross-hairs  will  then  be  made  visible  by  light 
diffused  by  the  tracing  cloth. 

For  a  few  minutes  just  before  dark  Polaris  can  be  easily  seen  through 
tne  telescope  before  it  can  be  seen  with  the  unaided  eye,  and  the  cross- 
hairs will  be  visible  against  the  sky  without  artificial  illumination.  In 
order  to  find  Polaris  under  these  circumstances  it  is  necessary  to  know  its 
approximate  altitude  at  the  time.  The  telescope  must  be  focussed  on 
some  very  distant  object  and  then  raised  until  the  vernier  indicates  the 
star's  altitude.  By  pointing  the  telescope  about  north  and  then  moving 
the  instrument  very  slowly  right  and  left  the  star  can  be  found.  For  the 
method  of  finding  the  altitude  of  the  star  see  Arts.  19  to  21,  and  equa- 
tion [7],  p.  29.  Observations  on  Polaris  just  at  dusk  can  be  utilized  when 
making  observations  by  the  method  of  Art.  26. 

If  accurate  results  are  desired  in  the  observations  for  azimuth  the 
instrument  should  be  firmly  set  up  and  allowed  to  stand  for  some  time 
before  the  observations  are  begun;  the  observations  should  be  made  as 
quickly  as  is  consistent  with  careful  work,  as  delay  simply  allows  the 
instrument  more  opportunity  to  change  its  position  and  thus  introduce 


CONVERGENCE  OF  MERIDIANS  5 

error.  Unless  the  transit  is  in  perfect  adjustment  it  is  well  to  make  two 
observations,  one  with  the  telescope  direct  and  the  other  with  the  telescope 
reversed,  and  to  use  the  mean  result. 

ii.  Azimuth  Mark.  —  In  finding  the  azimuth  of  a  transit  line  by  star 
observations  the  instrument  is  set  up  at  one  end  of  the  line,  and  at  the 
other  end  is  placed  some  sort  of  azimuth  mark  on  which  pointings  can 
be  made.  If  it  is  inconvenient  or  impossible  to  set  this  mark  at  the  other 
end  of  the  transit  line  it  may  be  set  in  any  convenient  position,  not  too 
near  the  instrument,  and  its  azimuth  from  the  instrument  determined. 
This  observed  direction  is  then  connected  with  the  survey  by  means  of 
an  angle  measured  in  the  daytime  between  the  line  to  the  azimuth  mark 
and  the  transit  line.  In  this  case  the  mark  should  be  so  arranged  that 
it  can  be  sighted  on  either  in  the  daytime  or  at  night.  For  an  accurate 
determination  of  the  azimuth  the  mark  usually  consists  of  a  box  with  a 
small  hole  cut  in  the  side  toward  the  observer  so  that  light  from  a  lantern 
placed  inside  can  shine  through  the  opening.  The  diameter  of  the  hole 
should  be  such  as  to  subtend  an  angle  of  about  one  second  (0.3  inch  per 
mile).  The  light  should  have  the  appearance  of  a  small  point  of  light 
like  a  star;  it  should  not  appear  large  or  blurred.  If  the  line  to  be  sighted 
over  is  not  one  of  the  lines  of  the  survey  but  is  to  be  connected  with  the 
survey  by  an  angle  measured  in  the  daytime,  the  box  should  have  a  target 
or  a  stripe  painted  on  it  to  serve  as  a  mark  when  sighting  on  it  in  the 
daytime.  The  centre  of  the  hole  and  the  centre  of  the  target  should 
coincide  and  should  be  placed  carefully  on  the  line  to  be  sighted  over. 
For  the  best  results  the  mark  should  be  placed  far  enough  away  so  that 
the  focus  of  the  telescope  does  not  have  to  be  altered  when  changing  from 
the  star  to  the  mark. 


\ 
FIG.  i.     Convergence  of  the  Meridians. 

12.  Convergence  of  Meridians.  —  In  comparing  azimuth  observations 
made  at  points  in  different  longitudes  it  will  be  necessary  to  allow  for 
the  angular  convergence  of  the  meridians  at  the  two  places.  In  running 


6  AZIMUTH 

westward,  in  the  northern  hemisphere,  the  meridians  are  turned  farther 
toward  the  right,  as  shown  in  Fig.  i. 

If  an  azimuth  observation  were  made  at  A  and  a  traverse  run  west- 
ward to  B  and  another  azimuth  observation  made  at  point  B,  then  it 
would  be  necessary  to  add  to  the  azimuth  observed  at  B  the  correction 
c  in  order  to  reduce  this  azimuth  to  what  it  would  be  if  referred  to  the 
meridian  at  A.  The  difference  between  this  corrected  azimuth  and  the 
azimuth  computed  from  the  angles  represents  the  accumulated  error  of 
the  angular  measurements  of  the  survey.  The  amount  of  this  correction 
for  convergence  of  the  meridians  is  shown  in  Table  II. 

METHODS    OF    OBSERVING. 

13.  Azimuth  from  an  Observed  Altitude  of  the  Sun.  —  In  the  following 
paragraphs  it  is  assumed  that  the  instrument  is  set  up  at  some  regular 
transit  station  of  a  survey  and  that  it  can  be  sighted  at  another  station. 
The  lower  motion  of  the  transit  should  remain  clamped  during  the  obser- 
vation. The  telescope  is  pointed  at  the  sun,  and  the  sun's  image  "found" 
in  the  field  and  sharply  focussed.  When  a  dark  glass  is  used  the  cross- 
hairs usually  cannot  be  seen  except  when  they  appear  against  the  sun's 
disc.  The  telescope  should  be  moved  up  and  down  a  little  so  that  the 
cross-hairs  and  also  the  stadia  hairs  can  be  identified  (against  the  sun's 
disc)  in  order  to  avoid  observing  on  a  wrong  hair.  The  observation  is 
made  by  measuring  the  altitude  of  the  upper  and  lower  edges  of  the  sun's 
disc  and  measuring  the  horizontal  angles  to  the  right  and  left  edges  so 
that  the  mean  of  these  pairs  of  observations  gives  the  altitude  of  the  sun's 
centre  and  a  horizontal  angle  to  the  centre  corresponding  to  the  same 
instant.  In  the  first  half  of  the  observation  the  vertical  cross-hair  is  set 
tangent  to  the  left  edge  of  the  sun  (by  means  of  the  upper  plate  tangent 
screw),  and  the  horizontal  cross-hair  is  set  tangent  to  the  upper  edge  of 
the  sun;  the  vertical  arc  and  the  horizontal  circle  are  then  read  and 
recorded.  The  watch  time  of  the  observation  is  also  noted.  The 
accuracy  of  the  result  may  be  increased  by  taking  several  such  pointings 
and  using  the  mean.  In  the  second  half  of  the  observation  the  vertical 
cross-hair  is  set  tangent  to  the  right  edge  and  the  horizontal  cross-hair 
tangent  to  the  lower  edge,  thus  placing  the  sun  in  the  opposite  quarter  of 
the  field  from  that  used  before.  Both  angles  and  the  time  are  again 
recorded.  The  same  number  of  pointings  should  be  made  in  the  second 
half  as  in  the  first.  The  telescope  should  then  be  levelled  and  the  vernier 
examined  to  see  if  there  is  any  index  correction  to  be  applied  to  the 
readings  of  the  vertical  arc.  If  the  plate  is  set  so  that  the  vernier  reads 
azimuths,  as  is  customary  in  stadia  surveying,  then  these  vernier  readings 


AZIMUTH  FROM  AN  OBSERVED  ALTITUDE  OF  THE  SUN       7 

alone  will  give  the  horizontal  angle  between  the  sun  and  the  meridian 
from  which  the  azimuths  are  being  read.  If  the  circle  is  not  set  for 
azimuths  the  upper  clamp  should  be  loosened  and  the  telescope  sighted 
on  some  line  of  the  survey  and  the  vernier  read  again,  so  that  this  vernier 
reading,  combined  with  the  two  readings  on  the  sun?  will  give  the  horizontal 
angle  between  the  sun  and  the  station  sighted. 

Theoretically  it  is  immaterial  whether  the  observations  are  made  in 
the  exact  order  given  above  or  not,  provided  the  sun  is  observed  first  in 
one  of  the  quadrants  formed  by  the  cross-hairs  and  then  in  the  opposite 
quadrant.  It  will  be  found,  however,  that  the  sun  moves  so  rapidly  that 
it  is  difficult  to  set  both  cross-hairs  accurately  in  position  at  the  same 
instant,  hence  the  observation  will  be  easier  and  also  more  accurate  if  we 
select  that  pair  of  opposite  quadrants  in  which  it  will  be  necessary  to 
make  but  one  setting  with  the  tangent  screw,  the  other  setting  being  made 
by  the  motion  of  the  sun  itself.  This  may  be  done  in  the  following 
manner.  If  the  observation  is  to  be  made  in  the  forenoon  set  the  vertical 
cross-hair  a  little  in  advance  of  the  left  edge  of  the  sun.  (see  the  lower 


FIG.  2a. 

A,M.  Observations. 


FlG.  2b. 
P.M.  Observations. 


Diagram  showing  position  of  sun's  disc  a  few  seconds  before  the 
instant  of  observation  for  azimuth  (northern  hemisphere). 

The  arrows  show  the  direction  of  the  sun's  motion.     Stadia  hairs  are  shown 
as  dotted  lines. 

part  of  Fig.  2a),  and  keep  the  horizontal  cross-hair  tangent  to  the  upper 
edge  of  the  sun  by  means  of  the  vertical  tangent  screw,  following  it  until 
the  left  edge  of  the  sun  has  moved  up  to  the  vertical  cross-hair.  At  this 
instant  stop  moving  the  tangent  screw,  note  the  time,  and  read  the  angles. 
For  the  right  and  lower  edges  the  position  is  as  shown  in  the  upper  part 
of  Fig.  2a,  the  horizontal  cross-hair  cutting  across  the  lower  portion  of 


AZIMUTH 

the  sun  and  the  vertical  cross-hair  being  kept  tangent  to  the  right  edge  by 
means  of  the  plate  tangent  screw.  For  observations  in  the  afternoon  the 
positions  will  be  as  shown  in  Fig.  2b.  If  a  transit  with  an  inverting 
eyepiece  is  being  used  these  positions  will  of  course  appear  reversed. 

If  the  transit  is  provided  with  stadia  hairs  care  should  be  taken  not  to 
mistake  one  of  them  for  the  middle  cross-hair.  In  the  figure  the  stadia 
hairs  are  represented  by  dotted  lines,  the  distance  between  them  being 
i/ioo  part  of  the  focal  length  of  the  objective.  The  angular  distance 
between  them  is  therefore  about  o°  34',  a  little  greater  than  the  diameter 
of  the  sun. 

If  the  transit  has  a  complete  vertical  circle  the  telescope  should  be 
reversed  between  the  two  pointings  on  the  sun  to  eliminate  errors  of 
adjustment. 

Repeated  trials  indicate  that  if  the  sun's  disc  is  bisected  with  both 
cross-hairs  at  the  same  instant,  the  error  in  the  resulting  azimuth  usually 
will  not  exceed  i'.  For  rapid  work  where  great  accuracy  is  not  de- 
manded this  method  may  be  used,  as  it  saves  time  and  removes  all 
doubt  as  to  which  limb  or  which  cross-hair  was  observed. 

14.  Computing  the  Azimuth  from  Sun  Observations.  —  In  order  to 
compute  the  azimuth  we  proceed  as  follows  : 

i.)  ALTITUDE.  — Take  the  mean  of  the  altitudes  of  the  upper  and 
lower  edges  of  the  sun,  and  subtract  from  it  the  refraction  correction  taken 
from  Table  I,  thus  obtaining  the  true  altitude.  The  index  correction 
must  also  be  applied. 

2.)  HORIZONTAL  ANGLE.  —  Take  the  mean  of  the  vernier  readings 
of  the  horizontal  circle  for  the  pointings  on  the  sun. 

If  a  pointing  has  been  made  on  some  reference  mark,  take  the  difference 
between  this  vernier  reading  and  the  mean  vernier  reading  for  the  sun, 
the  result  being  the  horizontal  angle  between  the  mark  and  the  sun. 

3.)  DECLINATION. — Take  from  the  Nautical  Almanac*  (or  any  solar 
ephemeris)  the  declination  f  of  the  sun  at  Greenwich  Mean  Noon  (G.M.N.) 
for  the  date  of  the  observation,  and  also  the  difference  for  i  hour  and  its 
algebraic  sign,  found  in  the  next  column  to  the  right.  In  order  to  obtain 
the  declination  at  the  instant  of  the  observation  it  will  be  necessary  to 
allow  for  the  change  in  the  declination  since  the  instant  of  Greenwich 
Mean  Noon.  If  the  watch  keeps  Standard  Time  the  correction  for 

*  It  is  not  necessary  to  use  the  large  Nautical  Almanac  for  obtaining  the  sun's  declina- 
tion. Pamphlets  containing  the  declination  and  the  equation  of  time  are  issued  by  the 
Hydrographic  Office  (Publication  No.  118)  and  may  be  obtained  from  the  regular  agents. 
Copies  of  the  solar  ephemeris  are  also  published  in  the  form  of  handbooks  for  engineers 
and  for  the  use  of  navigators.  Values  of  the  sun's  declination  for  the  years  1916  to  1919 
inclusive  will  be  found  in  Table  XV. 

fThis  is  given  in  the  Almanac  under  the  heading  Apparent  Declination. 


AZIMUTH  FROM  AN  OBSERVED  ALTITUDE  OF  THE  SUN    9 

change  in  declination  can  be  made  in  a  very  simple  manner.  At  the 
instant  of  Greenwich  Mean  Noon  it  is  7  A.M.  Eastern  Time,  6  A.M.  Central 
Time,  5  A.M.  Mountain  Time,  and  4  A.M.  Pacific  Time.  Hence  we  can 
obtain  the  time  elapsed  since  G.  M.  N.  by  simply  subtrapting  7  A.M., 
6  A.M.,  etc.,  as  the  case  may  be,  from  the  observed  watch  time,  first  adding 
12^  to  the  time  if  it  is  afternoon.  The  difference  for  i  hour  taken  from 
the  Almanac  is  to  be  multiplied  by  this  elapsed  time  expressed  in  hours 
and  the  result  added  to  or  subtracted  from  the  declination  at  G.  M.  N. 
(This  multiplication  may  be  avoided  by  the  use  of  Table  XIV.)  An 
examination  of  the  declination  for  the  preceding  or  following  dates  will 
show  whether  it  is  increasing  or  decreasing  and  hence  show  whether  the 
correction  is  to  be  added  or  subtracted.  If  the  declination  is  considered 
positive  when  the  sun  is  north  of  the  equator  and  negative  when  south, 
then  the  elapsed  time  multiplied  by  the  difference  for  i  hour  as  given 
in  the  Almanac  is  always  to  be  added.  For  example,  suppose  the  declina- 
tion is  desired  for  Nov.  10,  1909,  at  2h  30™  P.M.,  Eastern  time.  The  decl. 
for  G.  M.  N.,  Nov.  10,  =  —  17°  03'.!,  the  diff.  for  i  hour  =—  42".$. 
The  time  elapsed  since  G.  M.  N.  =  2*1  30™  P.M.  +  12^  —  7  A.M. 
=  14^  30™—  7*1  =  7^.5.  The  total  change  is  —  42". 5X7^.5  =  —  s'.3- 
The  corrected  declination  is  -  17°  03'.  i  -  5^.3  — —  17°  o8'.4.  The 
hourly  change  never  exceeds  i  minute  of  angle,  so  that  if  the  watch  is  in 
error  by  as  much  as  10  minutes  the  resulting  error  in  the  declination  will 
have  a  small  effect  on  the  azimuth. 

If  the  watch  keeps  local  time  the  watch  time  of  G.  M.  N.  is  found 
by  subtracting  from  12^  the  west  longitude  of  the  place  expressed  in 
hours,  minutes,  and  seconds.  For  example,  if  the  observer  were  in 
longitude  93°  W.  his  (local)  noon  would  occur  6h  i2m  after  G.  M.  N., 
i.e.,  5*1  48m  A.M.  by  his  watch  (if  correct)  is  the  instant  of  G.  M.  N. 

4.)  AZIMUTH.  —  Compute  the  azimuth  of  the  sun,  from  the  south  point, 
by  the  formula 

log  vers  Az.  =  log  [sin  \  go0-  (Lat.  +  Alt.)  J  +sin  Decl.] 

+  log  sec  Lat. 

+  log  sec  Alt.,       [i] 

in  which  Az.  is  the  azimuth  of  the  sun's  centre  from  the  south  point,  east 
or  west;  Lat.  is  the  latitude  of  the  place  either  taken  from  a  map  to  the 
nearest  minute  or  obtained  by  observation  (see  Art.  17);  Alt.  is  the  cor- 
rected altitude  of  the  sun's  centre;  and  Decl.  is  the  declination  of  the 
sun  at  the  instant  of  the  observation.  The  arrangement  of  the  compu- 
tation is  as  shown  in  Examples  i  to  3,  pp.  n  and  12.  The  latitude  and 
the  altitude  are  written  down,  their  sum  taken,  and  its  complement 


io  AZIMUTH 

written  beneath.  From  Table  III  we  take  the  log  secants  of  the  latitude 
and  the  altitude.  These  are  found  by  looking  up  the  secant  for  the  next 
smaller  angle  in  the  left  portion  of  the  table  and  adding  the  proportional 
parts  for  the  minutes  from  the  proper  column  at  the  right.  The  log 
secant  can  thus  be  written  down  directly,  to  the  nearest  unit  in  the  fourth 
figure.*  The  characteristics  (o  for  all  the  log  secants  occurring  within 
the  limits  of  this  table)  have  been  omitted  in  Table  III.  From  Table  IV 
we  obtain  the  natural  sine  of  90°— (latitude +altitude)f  and  also  nat- 
ural sin  declination  and  take  their  algebraic  sum.  If  the  declination  is  — 
the  sine  is  — .  From  Table  V  we  look  up  the  log  of  this  sum  and  add  it 
to  the  two  log  secants.  This  sum  is  the  log  vers  of  the  azimuth  reckoned 
from  the  south  point.  In  Table  VI  are  given  the  log  versed  sines,  the 
arrangement  being  exactly  as  in  the  preceding  tables.  If  the  observa- 
tion is  made  in  the  afternoon  the  angle  from  Table  VI  is  the  azimuth 
desired;  if  in  the  forenoon  the  angle  must  be  subtracted  from  360  degrees, 
since  azimuths  are  reckoned  from  the  south  point  in  a  clockwise  direction. 
This  azimuth,  combined  with  the  measured  horizontal  angle,  will  give  the 
azimuth  of  the  line  desired. 

If  it  is  desired  to  compute  the  azimuth  with  as  great  precision  as  the 
observations  will  afford,  i.e.,  to  about  5  or  io  seconds,  tables  carried  to 
five  places  should  be  used  in  the  computation. 

The  formula  given  above  applies  to  the  northern  hemisphere,  but  if  the 
algebraic  sign  of  the  declination  is  changed  and  the  azimuth  reckoned 
from  the  north  point  instead  of  the  south  it  will  apply  to  the  southern 
hemisphere.  In  the  southern  hemisphere  the  positions  of  the  sun  shown 
in  Fig.  i  will  obviously  be  changed. 

15.  When  to  Observe.  —  The  most  favorable  times  for  accurate  obser- 
vations by  this  method  are  when  the  sun  is  nearly  east  or  west.  If  the 
best  results  are  desired  the  observations  should  not  be  made  within 
2  hours  of  noon  nor  when  the  sun's  altitude  is  much  less  than  io 
degrees. 

*  The  table  extends  only  to  60  degrees,  which  is  sufficient  for  all  ordinary  cases,  f  hould 
it  be  necessary  to  find  the  log  secant  of  an  angle  greater  than  60  degrees  it  may  be  done 
by  taking  the  natural  sine  of  the  complement  (Table  IV),  looking  up  its  log  (Table  V)  and 
subtracting  this  log  from  zero.  For  angles  greater  than  about  80  degrees,  however,  this 
method  is  not  sufficiently  accurate. 

t  The  sine  is  employed  rather  than  cos  (latitude  +  altitude)  so  that  aH  numbers  may 
be  taken  from  the  tables  in  exactly  the  same  manner.  If  the  sum  of  the  latitude  and  the 
altitude  exceeds  90  degrees  the  natural  sine  of  this  angle  is  negative.  (See  Example  2 .) 


WHEN  TO  OBSERVE 


EXAMPLE  i. 


Latitude  42°  21'  N. 


OBSERVATION  ON  THE  SUN  FOR  AZIMUTH. 


Instrument  at  Sta.  no. 


Point  sighted 

Hor.  Circle 

Vert.  Ar 

Station  in 

238°  14' 

oj 

311    48 

14°  4i 

0.1 

312      20 

15   oo 

fo 

312      27 

15    55 

ro 

312      51 

16   08 

Mean 


Horizontal  Angle 


312°  21/5      Obs.Alt.  15°   26' 

238°  14 

74°  07' .5  Refr.  3.5 

True  Alt.  15°    2 2'. 5 


Nov.  28,  1905. 

Watch 

(Eastern  Time) 
8h  4im  A.M. 

8    47 
8h44« 


ih  44*  Gr.  Time 


COMPUTATION. 

Lat.  42°  21'     log  sec  .1313 

Alt.  15    22.5  log  sec  .0158     Sun's  Decl.at  G.  M.  N.  =  —  21°  14' .9 

Diff.i»  =  -  26"  .81 

Sum  57    43  -5 

„  o    *,  26" -8l  X  ih-73  =  -  o'.8 

Nat.  sin      .5340          Co      32°  i6'-s 

Nat.  sin  -  .3626      Decl.  -  21°  15' -7  Decl.  at  8"  44m  =  -  ai°  15' -7 

Alg.  Sum    .1714  log  9.2340 

log  vers  9.3811 
Az.  of  Sun    40°  34'.? 
Sta.  in  N.  of  Sun    74°  07' .5* 

114°  42' .2 
Az.  245°  18' 
Sta.  no  to  Sta.  in,  N.  65°  18'  E. 

*  The  plate  readings  (azimuths)  indicate  that  Sta.  in  was  to  the  left  of  the  sun  and 
hence  North  of  it. 


12 


AZIMUTH 


EXAMPLE  2. 

Latitude  42°  30'  N. 


AZIMUTH  OBSERVATION  ON  SUN. 


Instrument  at  Sta.  B. 


Hor.  Circle 

Vert.  Arc 

Sta.  7 

o°oo' 

|o 

99°  19' 

56°  35' 

12 

99°  54' 

56°  54' 

ol 

99°  4o' 

57°  49' 

ol 

100°  07' 

58°  02' 

Mean 

"99^4? 

57°  20' 

Refr.                0.7 

57°  19' -3 


July  15,  1907. 

Eastern  Time 
9h  47m  A.M. 
9    Si 
9h49» 


Decl.  at  G.  M.  N.  +  21°  40'.? 
22" .8  X  2h.8  =  -  i'.o 


Decl.  =  +  21°  39' .7 


COMPUTATION. 

Lat.  42°  30'     log  sec  .1324 
Alt.  57    19.3  log  sec  .2677 

99°  49' -3 

sin—  .1706  Co    —9°  49' -3 

sin       -3691         Decl.  +  21°  39'.? 

sum     .1985  log  9-2978 

log  vers  9-6979 
Az.  S.  59°  55'  E. 
Sta.  7  north  of  sun    99  45 

S.  159°  40'  E. 
Sta.  B  to  Sta.  7  =  200°  20' 

The  azimuth  of  Sta.  B  to  Sta.  7  as  calculated  from  the  angles  of  the  survey  is  200° 


EXAMPLE  3. 

AZIMUTH  OBSERVATION  AT  Q    25,  Aug.  6,  1907,  5ho4mP.M.  in  latitude  42°  29' .2  N. 
Mean  Alt.  =  22°  29' .3.     Mean  plate  reading  =  92°  35'  (supposed  to  be  true  azimuth). 


Decl.  at  G.  M.  N.  +  16°  57' .6 
40". 7  X  ioh.i  =  —  6'. 9 


Decl. 


+  16°  50'.? 


COMPUTATION. 

Lat.  42°  29' .2 
Alt.  22°  29  .3 

Sum   64°  58'  .5 

nat.  sin  .4230     Co         25°oi'.s  ' 
nat.  sin  .2898  Decl.  +  16°  50' .7 

Sum  .7128 


Hence  the  azimuths  read  at  Q  25  are  4'  too  small. 


log  sec    .1323 
log  sec    0343 


log  9-853° 

log  vers  0.0196 
Azimuth  92°  39' 
Vernier  92  35 

Error  =  04' 


LATITUDE  BY  THE  SUN  AT  NOON  13 

If  the  azimuth  calculated  by  the  preceding  rule  exceeds  70°  the 
azimuth  from  the  north  point  may  be  calculated  as  follows:  Take  the 
difference  between  the  Latitude  and  the  Altitude,  and  subtract  it 
from  90°.  From  the  natural  sin  of  this  angle  subtract  the  natural  sin 
Declination.  The  log  of  the  result  added  to  the  log  secants  of  the 
Latitude  and  Altitude  gives  the  log  vers  Az.  measured  from  the  north. 
This  affords  a  convenient  means  of  checking  azimuths  between  70°  and 
110°.  It  will  sometimes  be  found  that  the  two  azimuths  will  differ 
i',  or  even  2',  owing  to  the  fact  that  only  four  places  are  used  in  the 
logarithms.  The  mean  of  the  two  results  will  always  be  more  accurate 
than  the  result  of  a  single  computation. 

16.  Azimuth  by  Altitude  of  a  Star.  —  The  azimuth  of  a  star  can  be 
determined  in  the  same  way  as  the  azimuth  of  the  sun,  provided  the  star 
can   be  identified  and    its  declination  obtained.     Since  a  star  has  no 
appreciable  diameter  its  image  should  be  bisected  with  both  cross-hairs. 
Any  of  the  brighter  stars  contained  in  the  list  given  in  the  Nautical 
Almanac*  can  be  used  for  this  observation.     For  accurate  results  the 
star's  declination  should  not  be  greater  than  about  +  20  degrees  nor  less 
than  about  —  20  degrees,  and  at  the  time  of  the  observation  the  star 
should  be  nearly  due  east  or  due  west.     Since  the  declination  of  a  fixed 
star  does  not  change  appreciably  in  24  hours  it  will  not  be  necessary  to  note 
the  time  as  in  a  solar  observation. 

17.  Observation  for  Latitude.  —  In  order  to  obtain  the  sun's  azimuth 
it  is  generally  necessary  to  know  the  latitude  of  the  place  within  about 
i  minute.     In  many  cases  this  can  be  scaled  from  some  reliable  map  with 
sufficient  accuracy.     If  no  such  map  is  available  the  latitude  must  be 
observed  directly,  either  by  the  sun's  altitude  at  noon  or  by  the  altitude 
of  the  pole-star.     If  we  can  find  the  distance  of  the  point  of  observation 
north  or  south  of  some  other  point  whose  latitude  is  known,  the  latitude 
of  the  instrument  may  be  found  by  taking  6080  feet  equal  to  i  minute 
of  latitude.     The  latitude  can  often  be  found  in  this  way  in  places  which 
have  been  surveyed  by  the  United  States  Public  Lands  System,  since  the 
latitude  of  some  of  the  points   can  be  ascertained   and  the  distance 
north  or  south  to  other  points  found  from   the  township  and  section 
numbers. 

1 8.  Latitude    by    the    Sun   at   Noon.  —  This    observation    is    made 
by  measuring   the  altitude  of   the   sun  at  noon,    when   it   is  a   maxi- 
mum.    The  transit  should  be  set  up   and    levelled  some  time  before 


*  For  a  condensed  list  see  table  of  Fixed  Stars  in  the  Nautical  Almanac  under  the 
headimg  "Mean  Places";  for  the  exact  places,  right  ascension  and  declination  for  any  date, 
see  table  of  "  Apparent  Places."  A  short  list  is  given  on  p.  28  of  this  book. 


14  AZIMUTH 

noon*  and  the  horizontal  cross-hair  set  on  the  sun's  lower  edge.  As 
long  as  the  sun  continues  to  rise  it  should  be  followed  with  the  vertical 
motion  of  the  telescope,  keeping  the  cross-hair  exactly  tangent  to  the 
lower  edge  of  the  disc.  As  soon  as  the  sun  begins  to  drop  below  the  cross- 
hair the  motion  of  the  tangent  screw  should  be  stopped  and  the  verti- 
cal arc  read. 

This  altitude  must  be  corrected  for  (i)  index  error,  (2)  refraction, 
(3)  semi-diameter  of  the  sun,  and  (4)  the  sun's  declination.  The  refrac- 
tion may  be  taken  from  Table  I.  The  sun's  semi-diameter  may  be  taken 
from  the  Nautical  Almanac,  but  for  approximate  results  may  be  taken 
as  16'  in  March  and  in  September,  16'  15"  in  December,  and  15'  45" 
in  June.  If  the  lower  edge  of  the  sun  is  observed  the  correction  is 
to  be  added  to  the  measured  altitude;  the  upper  edge  could  have  been 
observed,  in  which  case  the  semi-diameter  should  be  subtracted.  The 
declination  of  the  sun  is  found  as  described  in  Art.  14,  or  it  may  be 
found  by  taking  from  the  Nautical  Almanac  the  declination  for  Greenwich 
Apparent  Noon,  multiplying  the  difference  for  i  hour  by  the  number 
of  hours  in  the  longitude  and  adding  this  to  the  declination  at  Greenwich 
Apparent  Noon.  The  declination  must  be  subtracted  from  the  altitude 
if  the  sun  is  north  of  the  equator  (+  ),  added  if  south  (—  ).  The  altitude 
thus  found  is  the  complement  of  the  latitude. 

EXAMPLE.  OBSERVATION  FOR  LATITUDE  BY  ALTITUDE  OF  Q  AT  NOON,  Jan.  13, 
1905.  Longitude  4h  4Sm  W.  (approx.).  Watch  time  =  nh  53™  (Eastern  Time). 

First  Method.  Second  Method 

Ded.  at  G.  M.  N.  =  —  21°  32' .6          Decl.  at  Apparent  Noon  =  —  21°  32'  31" 
25" .2  X  4b-9  =          +  2  .1  25" .2  X  4-75  =  +  2'  oo" 

Ded.  —  21°  30' .5          Deci.  at  Local  Noon  —  21°  30'  31" 

Maximum  Altit-de  O  25°  55' 
Refraction  2' 

25°  53' 
Semi-diameter        16  .3 

Altitude  of  centre  26°  09' .3 
Declination—  21    30  .5 

Complement  of  Latitude  47°  39'  A 
Latitude  42°  20' .2 

*  It  should  be  remembered  that  the  time  of  the  sun's  maximum  altitude  may  differ 
considerably  from  noon  by  the  watch.  To  obtain  the  Standard  Time  of  this  observation, 
call  the  time  of  the  observation  i2h,  Apparent  Time;  reduce  this  Apparent  Time  to  Mean 
Tune  by  adding  or  subtracting  the  equation  of  time  as  given  in  the  Nautical  Almanac;  then 
reduce  the  Mean  Time  to  Standard  Time  by  taking  the  difference  in  longitude  (expressed 
in  h.  m.  s.)  between  the  place  and  the  standard  meridian  and  adding  it  if  the  place  is  west 
of  the  standard  meridian,  subtracting  if  the  place  is  east. 


EXPLANATION  OF  METHOD  15 

19.  Azimuth    and    Latitude    by    Observation    of    3    Cassiopeiae    and 
Polaris.  —  The  method  described  in  the  following  articles  is  applicable 
when  only  approximate  results  are  desired,  say  within  about  i   minute 
of  the  true  values,  and  when  it  is  desired  to  obtain  the  result  quickly  and 
without  using  the  Nautical  Almanac.     An  advantage  offered  by  this 
method  is  that  it  is  not  necessary  to  know  the  local  time,  since  this  is 
determined  with  sufficient  accuracy  by  the  observation  itself.     Tables 
VII  to  XII  are  so  arranged  that  all  of  the  quantities  needed  in  this  obser- 
vation may  be  found  by  interpolation,  and  usually  all  the  tables  required 
for  an  observation  appear  at  the  same  opening  of  the  book. 

20.  Finding  the   Stars.  —  In   order   to   make  this   observation   it   is 
necessary  to  be  able  to  identify  certain  stars  near  the  north  pole.     The 
most  conspicuous  constellation  in  the  northern  sky  is  the  Great  Dipper, 
or  Great  Bear  (Ursa  Major}.     (See  Fig.  3.)     Polaris,  the  star  on  which 
the  azimuth  observation  chiefly  depends,  is  readily  found  by  reference 
to  the  Great  Dipper.     The  two  stars  forming  the  side  of  the  dipper  bowl 
which  is  farthest  from  the  handle  are  called  the  pointers  because  a  line 
through  them  points  very  nearly  to  Polaris,  the  distance  to  Polaris  being 
about  five  times  the  distance  between  the  pointers.      On  the  opposite  side 
of  the  pole  from  the  Great  Dipper  is  the  constellation  Cassiopeia,  shaped 
like  a  letter  W.     The  star  d  (delta)  Cassiopeiae,  which  is  to  be  used  in 
this  observation,  is  the  one  at  the  bottom  of  the  first  stroke  of  the  W,  i.e., 
the  lower  left-hand  star  when  the  W  is  right  side  up.     The  Little  Dipper 
is  an  inconspicuous  constellation;  Polaris  is  at  the  end  of  the  dipper 
handle,  and  two  fairly  bright  stars  form  the  outer  side  of  the  dipper  bowl. 
The  other  stars  in  this  constellation  are  quite  faint.     Another  star  which 
wrill  be  referred  to  later  is  d  Draconis.     When  d  Cassiopeiae  is  above  the 
pole  d  Draconis  will  be  found  west  (left)  of  the  meridian  at  about  the 
same  altitude  as  Polaris,  these  three  stars  forming  a  right  triangle  (nearly), 
the  right-angle  being  at  Polaris.     The  distance  from  Polaris  to  d  Draconis 
is  less  than  the  distance  from  Polaris  to  d  Cassiopeiae.     It  will  be  observed 
that  d  Draconis,  Polaris,  and  the  lower  star  in  the  bowl  of  the  Little  Dipper 
form  a  triangle  which  is  nearly  equilateral,     d  Draconis  is  not  as  bright 
as  d  Cassiopeiae.     There  are  several  faint  stars  near  d  Draconis,  one  of 
which  might  possibly  be  confused  with  it.     This  other  star  (e  Draconis) 
is  nearly  on  a  line  drawn  from  d  Draconis  to  d  Cassiopeiae,  and  its  distance 
from  d  Draconis  is  about  4  degrees,  a  little  less  than  the  distance  between 
the  pointers,     s  Draconis  is  not  as  bright  as  d  Draconis. 

21.  Explanation  of  Method.  —  In  order  to  determine  the  azimuth  of 
Polaris  and  the  latitude  of  the  place  it  is  necessary  to  find  by  some  means 
the  position  of  Polaris  with  respect  to  the  pole  at  the  instant  of  the  obser- 
vation.    This  depends  upon  the  hour  angle  of  Polaris  at  the  time  of  the 


i6 


AZIMUTH 


(-aunp) 


•  X 

Great     /  ^ 

**C  /     V 

jp.  ^^f 


HORIZON 


Polaris  at  Upper  Culmination.     (December.) 

DIAGRAM    SHOWING   THE  CONSTELLATIONS  ABOUT  THE   NORTH   POLE. 
The  arrows  show  the  direction  of  the  apparent  motion  of  the  stars. 

FIG.  3. 


EXPLANATION  OF  METHOD  17 

observation.  In  order  to  determine  the  coordinates  of  Polaris,  i.e.,  its 
distance  above  or  below  the  pole  and  its  distance  east  or  west  of  the 
meridian,  we  measure  the  altitude  of  the  star  d  Cassiopeiae  and  also  the 
altitude  of  Polaris.  From  these  altitudes  we  can  calculate  the  coordinates 
of  Polaris.  By  referring  to  Fig.  4  it  will  be  seen  that  the  pole,  Polaris, 
and  d  Cassiopeiae  are  all  nearly  in  the  same  plane  (i.e.,  on  the  same  hour 
circle),  the  two  stars  being  on  the  same  side  of  the  pole.  The  direction  of 
the  apparent  motion  of  the  stars  is  shown  by  the  arrow.  Hence  the 
relative  position  of  d  Cassiopeiae  and  Polaris  as  seen  by  the  observer  is  at 
once  a  key  to  the  position  of  the  pole  itself.  If  d  Cassiopeiae  is  directly 
above  Polaris,  then  Polaris  is  above  the  pole  and  nearly  in  the  meridian; 
if  d  Cassiopeiae  is  below  and  to  the  left  of  Polaris,  the  latter  is  below  and 
to  the  left  of  the  pole.  We  may  think  of  these  two  stars,  then,  as  moving 
around  the  pole  together  as  though  they  were  two  points  on  an  arm 
pivoted  at  the  pole. 


\EA8T 


HORIZON 
FIG.  4. 

If  we  know  the  latitude  of  the  observer,  and  the  polar  distance  and  the 
altitude  of  d  Cassiopeiae  at  any  instant,  we  may  calculate  the  hour  angle 
of  this  star,  i.e.,  the  arc  MS  in  Fig.  4.  Hence  if  Polaris  were  exactly  on 
the  same  hour  circle  with  d  Cassiopeiae  the  hour  angle  of  Polaris  would 
be  the  same  as  that  computed  for  d  Cassiopeiae.  In  reality  Polaris  has 


:8 


AZIMUTH 


a  slightly  smaller  hour  angle  than  d  Cassiopeiae,  the  difference  between 
the  two  increasing  slowly  from  year  to  year.  This  interval  is  6m  583  for 
the  year  1910,  ic111  573  for  1920,  and  i5m  138  for  1930.  After  we  observe 
the  altitude  of  d  Cassiopeiae  we  may  wait  until  this  interval  of  time  has 
elapsed  and  then  make  the  observation  on  Polaris,  the  latter  then  being 
in  the  position  it  would  have  occupied  at  the  first  observation  if  the  two 
stars  were  on  the  same  hour  circle.  This  instant  at  which  Polaris  is  to 
be  observed  we  may  call  for  convenience  the  computed  time.  When  the 
hour  angle  of  Polaris  is  known  for  the  instant  of  the  observation  the 


(Polaris) 


FIG.  5.     Coordinates  of  Polaris. 


coordinates  may  be  found  at  once.     In  Fig.  5  if  p  is  the  polar  distance  of 
Polaris  and  /  its  hour  angle,  then 

PM=p  cos/ 
and  SM=psmt.  [2] 

PM  is  the  amount  (nearly)  which  Polaris  is  above  or  below  the  pole, 

hence 

Lat.*  =Alt.  of  pole  =Alt.  of  Polaris— p  cos  /.  [3] 

*  For  a  more  accurate  expression  for  the  latitude  we  should  add  to  the  above  series 
the  quantity  K,  from  Table  XIII.    See  equation  [7]. 


THE  TABLES  19 

If  cos  t  is  given  its  proper  algebraic  sign  in  the  different  quadrants  this 
equation  holds  true  for  all  positions  of  the  star.  The  quantities  PM 
and  SM  may  be  found  in  Tables  VIII  and  XI. 

The  coordinate  p  sin  t  is  the  angular  distance  of  the  star  east  or  west 
of  the  meridian.  The  azimuth  of  Polaris  depends  not  only  upon  the 
distance  p  sin  t  but  also  upon  the  altitude  of  the  star  as  seen  by  the  obser- 
ver; it  may  be  found  by  the  equation 

Azimuth  =p  sin  t  sec  altitude,  [4] 

or  Azimuth  —  p  sin  t  +  p  sin  /  exsec  altitude.*  [5] 

The  azimuth  is  computed  by  taking  from  Tables  IX  or  XII  the  azimuth 
correction  (p  sin  /  exsec  altitude)  and  adding  it  to  p  sin  /.  This  azimuth 
is,  of  course,  reckoned  from  the  north  point. 

When  d  Cassiopeiae  is  near  the  meridian,  i.e.,  nearly  above  or  nearly 
below  Polaris,  an  accurate  determination  of  its  hour  angle  cannot  be 
made.  In  this  case  another  star,  d  Draconis  (p.  15),  can  be  substituted 
for  d  Cassiopeiae  and  the  hour  angle  of  Polaris  derived  in  a  similar  manner, 
so  that  it  is  almost  always  possible  to  use  this  method. 

22.  The  Tables.  —  All  of  the  quantities  needed  in  this  observation  may 
be  taken  from  Tables  VII  to  XII.  In  Table  VII  are  given  the  hour 
angles  of  d  Cassiopeiae  for  different  altitudes  and  different  latitudes. 
The  hour  angle  for  any  latitude  and  any  altitude  at  which  an  accurate 
observation  can  be  made  may  be  found  by  interpolation  in  this  table. 
If  the  star  is  east  of  the  meridian  the  tabular  hour  angle  should  be  sub- 
tracted from  360  degrees  to  obtain  the  true  hour  angle.  Since  it  is 
necessary  to  interpolate  in  this  table  both  for  the  altitude  and  the  latitude 
it  will  be  simpler  and  also  more  accurate  to  observe  the  star  when  the 
altitude  is  a  whole  degree,  preferably  an  even  numbered  degree,  and  thus 
confine  the  interpolation  to  the  latitudes.  If  the  observation  on  Polaris 
follows  the  observation  on  d  Cassiopeiae  by  the  interval  of  time  corre- 
sponding to  the  date,  as  given  above,  then  the  hour  angle  of  Polaris  at 
the  time  it  was  observed  is  the  same  as  the  hour  angle  taken  from  Table 
VII.  In  Table  VIII  will  be  found  the  values  of  p  sin  /  and  p  cos  t  for 
different  hour  angles  of  Polaris  between  30  degrees  and  150  degrees,  and 
for  the  years  1910,  1920,  and  1930.  If  the  date  falls  between  those  given 
in  the  table  it  will  be  necessary  to  interpolate  to  obtain  the  coordinates 
for  the  date  of  the  observation.  With  the  value  of  p  sin  t  found  in  Table 
VIII  and  the  measured  altitude,  we  take  from-  Table  IX  the  correction 
p  sin  t  exsec  altitude.  This  correction,  added  to  p  sin  t,  gives  the  azimuth 
desired. 

*  The  external  secant,  or,  the  secant  minus  unity. 


20  AZIMUTH 

It  will  be  noticed  that  in  general  we  do  not  know  the  latitude  of  the 
place,  and  therefore  we  cannot  determine  the  hour  angle  in  a  direct 
manner  as  was  assumed  above.  If  the  latitude  is  not  known  we  may 
proceed  as  follows.  From  the  relative  position  of  d  Cassiopeise  and 
Polaris  we  estimate  the  latitude,  remembering  that  Polaris  is  i°  10'  from 
the  pole,  and  its  direction  from  the  pole  is  nearly  the  same  as  the  direction 
of  d  Cassiopeiae  from  Polaris.  With  this  approximate  latitude  and  the 
measured  altitude  we  take  from  Table  VII  an  approximate  hour  angle  of 
d  Cassiftpeiae.  With  this  approximate  hour  angle  we  take  from  Table 
VIII  the  value  of  p  cos  /.  This  is  the  correction  to  be  applied  to  the 
altitude  to  give  the  corrected  latitude.  (Equation  [3].)  With  this  new 
latitude  we  take  from  the  table  a  more  accurate  value  of  the  hour  angle. 
Using  this  new  hour  angle  we  find  from  Table  VIII  the  value  of  p  sin  t. 
Two  approximations  will  always  give  p  sin  /  with  sufficient  accuracy. 
The  azimuth  correction  from  Table  IX  may  be  taken  out  as  before.  If 
the  latitude  as  well  as  the  azimuth  is  desired  the  above  process  of  approxi- 
mation may  be  continued  if  necessary  until  the  value  of  p  cos  t  agrees 
within  about  i  minute  with  the  preceding  value.  The  latitude  is  then 
found  by  equation  [3]. 

It  is  not  really  necessary  to  observe  Polaris  at  the  same  hour  angle  as 
d  Cassiopeiae,  although  this  simplifies  the  calculation  slightly.  In  case 
the  observation  is  not  made  at  the  computed  time  we  must  correct  the 
hour  angle  accordingly  before  taking  out  p  cos  /  from  Table  VIII.  The 
correction  is  made  by  converting  this  difference  in  the  time  interval  into 
degrees  and  adding  it  to  or  subtracting  it  from  the  hour  angle  of  d  Cassio- 
peiae. Suppose  for  instance  that  the  observation  on  Polaris  was  made 
501  208  before  the  calculated  time,  then  5m  208  =  5m-33  =  i°-33  (see 
Art.  5),  which  must  be  subtracted  from  the  hour  angle  of  d  Cassiopeiae 
to  obtain  the  hour  angle  of  Polaris,  since  Polaris  had  not  reached  this  hour 
angle  at  the  instant  it  was  observed. 

23.  Making  the  Observations.  —  Set  up  the  instrument  at  one  end  of 
the  line  whose  azimuth  is  to  be  found,  and  set  a  lantern  or  arrange  an 
azimuth  mark  at  the  other  end  of  the  line.  See  if  d  Cassiopeiae  is  in  a 
favorable  position  for  an  observation.  If  d  Cassiopeiae  is  near  the  merid- 
ian, either  above  or  below  the  pole,  an  accurate  observation  cannot  be 
made  on  this  star.  If  the  altitude  of  the  star  and  the  latitude  of  the 
observer  are  such  that  an  hour  angle  can  be  found  in  Table  VII,  then  a 
reliable  observation  can  be  made.  If  it  is  found  that  d  Cassiopeiae  can 
be  used,  point  the  telescope  at  this  star,  examine  the  vertical  circle  to 
see  what  the  approximate  altitude  is,  and  set  it  so  that  the  vernier  reads 
a  whole  degree  (preferably  an  even  numbered  degree)  +  the  refraction 
correction  for  this  altitude  (Table  I).  If  the  star  is  west  of  the  meridian 


MAKING  THE  OBSERVATIONS  21 

it  is  moving  downward,  and  the  telescope  must  be  set  at  some  altitude 
below  that  of  the  star.  If  the  star  is  east  of  the  meridian  the  telescope 
must  bo  set  at  a  higher  altitude  than  that  of  the  star.  Watch  the  star 
and  note  the  time  when  it  crosses  the  horizontal  cross-hair.  The  star 
moves  so  slowly  that  the  observation  is  not  precise,  but  it  is  sufficiently 
exact  for  this  purpose.  Next  calculate  the  time  of  the  observation  to  be 
made  on  Polaris  by  adding  to  the  watch  time  just  noted  the  interval 
from  the  top  of  Table  VII.  Set  the  plate  vernier  at  o  degrees  and  point 
at  the  azimuth  mark,  using  the  lower  clamp.  Loosen  the  upper  clamp, 
point  the  telescope  toward  Polaris,  and  set  both  cross-hairs  on  the  star. 
Follow  the  star's  motion,  using  the  vertical  tangent  screw  and  the  upper 
plate  tangent  screw,  until  the  computed  time  is  indicated  by  the  watch, 
then  see  that  both  cross-hairs  are  bisecting  the  star.  Read  the  vertical 
arc  and  the  plate  vernier,  and  determine  the  index  correction  to  the 
vertical  angle.  The  latitude  and  the  azimuth  are  then  found  from  Tables 
VII  to  IX  as  described  on  pp.  19  and  20. 

The  method  of  using  these  tables  is  illustrated  by  the  following  examples. 
The  first  two  illustrate  observations  in  which  Polaris  was  observed  at  the 
calculated  instant.  In  Examples  3  and  4  Polaris  was  observed  before 
the  computed  time  arrived. 

EXAMPLE  i .  OBSERVATION  ON  POLARIS  AND  8  CASSIOPEI.*:  FOR  LATITUDE  AND  AZIMUTH. 

March  14,  1908. 

Set  telescope  at  altitude  26°  02'  (26°  +  the  refraction  correction  for  26°);  8  Cassiopeia 
passed  horizontal  cross-hair  at  ph  o6m  10".  Interval  for  1008  =  6m  iog  (Table  VII,  tcp). 
Computed  time  for  observation  on  Polaris  =  ph  12™  20".  Set  on  mark  with  vernier  at 
o°oo'.  Bisected  Polaris  with  both  cross-hairs  at  ph  12™  20*.  Altitude  =  41°  52'. 5. 
Index  correction  =  —  Q'.S.  Horizontal  angle,  38°  31'. 5.  Mark  is  west  of  Polaris. 
5  Cassiopeia^  is  below  and  west  of  Polaris. 

FIRST  APPROXIMATION. 

Approx.  Lat.  =  42°  Alt.  41°  52' .5 

True  Alt.  =  26°  Index  corr.        —  0.5 

From  Table  VII,  hour  angle,  /  =  ii2°.o  41°  52' .o 

Refr.       —  i.i 

True  Alt.  41°  50' .9 
From  Table  VIII,  for  112°,  we  find  p  cos  /  =  —  26' .6  p  cos  i     —  26.6 

Approx.  Lat.  42°  17'. 5 
SECOND  APPROXIMATION. 

From  Table  VII,  using  lat.  42°  if.  5,  t     =    112°  .7 
From  Table  VIII,  for  112° .7,  p  cos  /  =  —  27' .3 

and  p  sin  t    =      65' .6  Alt.  41°  50' .9 

From  Table  IX,  az.  corr.  =       22  .5  p  cos  /     —  27.3 

Az.  =      8&.i  =   i°  28'.!  Lat.  42°  i8'.a 

Measured  angle  =38°  31'. 5 

Azimuth  of  mark  is      N.  39°  59' .6  W. 


22  AZIMUTH 

EXAMPLE  2.  OBSERVATION  ON  POLARIS  AND  6  CASSIOPEIA.    August  18,  1908. 
Set  telescope  at  altitude  34°oi'.5    (34°  +  refraction  corr.).      8   Cassiopeise   passed 
horizontal   cross-hair  at  9ho7mn8.      Interval  for  1908=  6m  ios.      Calculated  time  for 
observation  on  Polaris  =  ph  13™  21*.      Set  on  mark  with  vernier  at  o°  oo'.      Set   on 
Polaris  at  gh  i3m  21".      Altitude  =41°  46'.      Index  correction  =  o.      Horizontal  angle 
=  89°  38'.     Mark  east  of  Polaris.     3  Cassiopeia?  below  and  east  of  Polaris. 
Approx.  Lat.  42° 

True  Alt.  34°  Alt  =      „    6, 

Table  VII,  hour  angle  =    92° -9  Refr  =         \,  x 

True  hour  angle  =  360°  —  92°. 9  =  267°. i  41°  44^9 

From  Table  VIII,  for  g2°.g,  p  cos  /  =  —  3^.6  p  cos  /  =        —  3.6 

Approx.  Lat.  =  41°  48'. 5 

Table  VII,  lat.  41°  48'.s,  /  =    92°.6  41°  44'. g 

Table  VIII,  for  92°.6,  p  cos  /  =  -  3'. 2  -  3/.2 

and  p  sin  /  =  71'  =  i°  n'  Lat.  =  41°  48'. i 

Table  IX,  az.  corr.  =  24  -2 


Sum  =  true  az.  = 
Measured  horizontal  angle  = 

Angle  from  North 
Bearing  of  mark 

I°35'.2 
8938 

91°  13'  2 

S.  88°  46'  8  E. 

EXAMPLE  3.  OBSERVATION  ON  POLARIS  AND  d  CASSIOPEIA.    Feb.  n,  1008. 

Set  telescope  at  altitude  55°  01'  (5 5°+ refraction  corr.).    d  Cassiopeia;  passed  horizontal 
cross-hair  at  7h  03™  05'.      Interval   for  1008  =  6m  io3.      Computed  time  of  observation 
on  Polaris  =  7ho9m  15".     At    7h  07™  05"  Polaris  was    bisected    with    both   cross-hairs. 
Altitude  =  43°  05' .    Index  correction  =  + 1 ' .    Angle  between  Polaris  and  mark  =  67°  1 1  '„ 
Mark  is  east  of  Polaris,     d  Cassiopeiae  is  above  and  west  of  Polaris. 
Estimated  latitude  =42° 
True  altitude  =  55° 
From  Table  VII,  /  =49°  .9 
Interval  =    0.5 

Hour  angle  of  Polaris  =49° -4 

From  Table  VIII,  p  cos  /  =  +  45'  .6  Computed  time  7  -  09  - 1 5 

Observed  time  7—07—05 
.'.  Latitude  =42°  19' .4 

Interval  =  2™  io» 

=  o°.S 

SECOND  APPROXIMATION. 

Corrected  hour  angle  =    50°  .4 

Interval  =  .5 

Hour  angle  of  Polaris  =    49°  .9 

Table  VIII,  p  sin  /=     54' 4 

Table  IX,  corr .  =      1 9 .9 

Azimuth  of  Polaris  =    74'  .3 

=    i°  14' 

Measured  angle  =67°  n' 

Direction  of  mark  =N.  65°  57'  E. 


OBSERVATIONS  ON  8  DRACONIS  23 

EXAMPLE  4.  OBSERVATION  ON  POLARIS  AND  d  CASSIOPKI^F..     Jan.  2,  1908. 
$  Cassiopeia?  —  Alt.  59°  1  1  '.5;   time,   9ho7mso3.     Interval,  6m  io8.     Computed  time, 
9h  14™  oo3.     Polaris  —  Alt.  43°  13';  time,  ob  nmoos.     Angle  from  mark  to  Polaris,  106° 
49';  mark  west  of  star,     d  Cassiopeiae  is  above  and  W.  of  Polaris. 

Assumed  latitude  =42°  Computed  time    9h  i4m  oo' 

True  altitude  =59°  1  1'  Observed  time    9    n     oo 

Table  VII,  /=  41°.  4 

Table  VIII,  p  cos  t  =  4-  53'  .4  Interval    =  3™  oo« 

.'.  Latitude  =42°  19'  (approx.) 

SECOND  APPROXIMATION. 

Table  VII,  corrected  /  =41°.  5 
Interval,  3m^=  _  .8_ 

Hour  angle  of  Polaris  =40°  .7  Alt.  43°  13'     ' 

Table  VIII,  p  sin  /  =  46'.3  Refr.          i' 

Table  IX,  corr.  =17.2  '     ~ 

__  1  _  43    12' 

63'  .5  P  cos  t  +53-3 

Az.  of  Polaris  =     i°  03'.  5  Lat.  42°  iS'.y 

Horizontal  angle  =  106°  49' 


Az.  of  mark  S.  72°  07'.  5  W. 

24.  Observations  on  d  Draconis.  —  When  d  Cassiopeiae  has  an  hour 
angle  of  less  than  30  degrees  or  more  than  150  degrees  the  hour  angle 
cannot  be  accurately  determined  from  the  measured  altitude,  and  con- 
sequently Table  VII  does  not  include  such  hour  angles.  In  this  case  the 
altitude  of  d  Draconis  may  be  observed,  since  this  star  is  nearly  always 
in  a  favorable  position  for  an  observation  at  times  when  d  Cassiopeiae  is 
in  an  unfavorable  position.  The  observation  is  made  in  just  the  same 
way  as  for  d  Cassiopeiae  except  for  the  time  interval  between  the  two 
observations.  The  difference  in  hour  angle  of  d  Draconis  and  Polaris 
is  6h  i4m  2is  for  1910  —  too  long  to  wait  —  hence  we  make  the  obser- 
vation on  Polaris  as  soon  as  convenient  after  the  altitude  of  d  Draconis 
has  been  measured,  and  correct  the  hour  angle  of  d  Draconis  as  previously 
described  for  d  Cassiopeiae.  For  example,  if  the  altitude  of  d  Draconis 
was  taken  at  8h  i5m  P.M.  in  the  year  1910,  adding  6h  i4m  2is  gives 
i^h  29111  2  is  as  the  calculated  time  when  Polaris  will  have  the  same  hour 
angle  as  d  Draconis.  If  the  observation  were  made  on  Polaris  at  8h  2om 
oos  P.M.  we  must  subtract  from  the  true  hour  angle  of  §  Draconis  the 
quantity  i4h  29m  2is  —  8h  2om  oo8  =  6h  09™  2is  =  920-3,  which  will  give 
the  hour  angle  of  Polaris  at  the  time  it  was  observed.  The  difference  in 
hour  angle  between  Polaris  and  d  Draconis  for  different  dates  is  as 
follows:  —  1910,  6h  1401  2is  =  93°.6;  1920,  6h  18™  58^  =  94°.;; 
1930,  6h  23m  548  =  96°.  o.  The  following  examples  will  illustrate  the 
method  6f  making  the  calculations. 


AZIMUTH 


EXAMPLE  5. 

Jan.  10,   1908. 

Set  telescope  at  Altitude  37°  01'.  d  Draconis  passed  horizontal  hair  at  sh  57m  to1. 
Interval  for  1008  =  6h  13™  26*.  Observed  Polaris  at  5h  59™  40';  altitude  =  43°  30' ; 
plate  vernier,  275°  2 7'. 5;  mark,  216°  13' .o.  d  Draconis  west  of  Polaris. 


Estimated  latitude=42° 
True  altitude=  37° 
Table  X,  Hour  Angle=93°-3 
Hour  Angle  of  Polaris*=93°.3—  92° .7=  +o°.6 
Table  XI,  p  cos  /=  + 1°  n 
:.  Latitude=42°  18' 
Table  X,  Corrected  Hour  Angle=Q4°.i 
Correction  for  interval  =g^_^_ 
Hour  Angle  of  Polaris  =   i°^ 


Observed  time    sh  57" 
Interval   6    13 


26 


Computed  time  12    10    36 
Observed  time    5    59    40 


Diff .    6h  iom  56" 
=  92°  .7 

Vernier  Readings. 
275°27/-S 
216  13 
Angle  =  59°  1 4' .5 


Table  XI,  p  sin  *=oi'.7 

Table  XII,  correction= .6_ 

Azimuth  Polaris=  02' .3 

Angle  to  mark=       59°  14' -5 
Azimuth  of  mark=N.  59°  i6'.»  W. 


Alt.  43°  30' 
Refr.  i' 


43°  29' 

P  cost  +i    ii 
Lat.  42°  1 8' 


EXAMPLE  6. 

Jan.  2,  1908. 

d  Draconis;  —  Observed  altitude  =  27°  25';  observed  time  =  8h  32™  45*.  Inter- 
val =  6h  13™  26*;  computed  time=i4h  46™  n1;  Polaris;  —  altitude  =  43°  20';  observed 
time=8h  34™  45'.  Angle,  mark  to  Polaris,  107°  01'.  Mark  is  west  of  Polaris.  d  Dra- 
conis is  west  of  Polaris. 

Assumed  latitude=42° 
True  altitude=  27°  23' 
Table  X,  /=  124°  JQ 


Computed  time=i4h  46™  ii§ 

Observed  time=   8    34    45 

Interval  =   6    n 


26 


Hour  angle  of  Polaris=i24°.o-92°.8=3i0.2 


=  92°  .8 


Table  VIII  for  31°  .2,  p  cos  /=  +60' 

.*.  Latitude=42°  19'  (approx.) 
Table  X,  corrected  /=i25°.2 
Hour  angle  of  Polaris=32°.4 


Table  VIII,  p  sin  t= 
Table  IX,  correction= 


38'.o 
14  a 


Corrected  altitude  =  43°  19 
P  cos /=  4-i°  oo' 

Latitude  =  42°  19' 


Azimuth  of  Polaris=  52'.2 

Angle=    107°  01' 


Azimuth  of  mark= 


107    53-2 
=  S.  72°  07'  W. 


*  In  case  this  Hour  Angle  becomes  negative  it  should  be  subtracted  from  360  degrees. 
Polaris  in  that  case  would  be  east  of  the  meridian. 


MERIDIAN  LINE  BY  POLARIS  AT  CULMINATION 


25 


The  above  examples  have  been  worked  out  more  elaborately  than 
would  be  required  in  many  cases.  Frequently  the  latitude  will  be  known 
or  may  be  estimated  closely,  so  that  only  one  approximation  is  needed. 
Nearly  all  of  the  interpolation  may  be  done  mentally. 

25.  Meridian  Line  by  Polaris  at  Culmination.  —  When  Polaris  is  near 
the  meridian  (i.e.  near  culmination)  it  will  sometimes  be  convenient  to 
use  the  following  simple  method,  which  will  give  the  meridian  with  about 
the  same  accuracy  as  the  method  of  Articles  19-24.  We  may  use  in 
this  case  either  o  Cassiopeiae  or  the  star  f  in  the  Great  Dipper  (see 
Fig.  3).  If  we  determine  by  means  of  a  surveyor's  transit  the  instant 
when  one  of  these  stars,  say  d  Cassiopeiae,  is  vertically  above  or  vertically 
below  Polaris,  then  we  have  only  to  wait  a  certain  interval  of  time,  depend- 
ing upon  the  date,  when  Polaris  will  be  in  the  meridian.  If  Polaris  is 
sighted  at  this  instant,  the  telescope  may  be  lowered  and  the  direction  of 
this  line  marked  on  the  ground,  thus  giving  a  meridian  line  without 
further  calculation  The  instant  when  the  two  stars  are  in  the  same 
vertical  plane  cannot  be  determined  precisely,  but  it  can  be  observed  with 
all  the  accuracy  required  by  setting  the  cross-hair  on  Polaris  a  few  minutes 
before  they  are  in  the  same  vertical  plane  and  then  noting  the  instant 
when  the  other  star  passes  the  vertical  cross -hair.  If  the  interval  between 
the  two  observations  is  not  more  than,  say,  5m,  Polaris  will  change  its 
direction  so  little  that  the  effect  on  the  observed  time  of  the  other  star 
may  be  neglected.  A  convenient  way  to  keep  this  interval  small  is  to  set 
on  Polaris,  lower  the  telescope,  and  wait  until  the  other  star  appears  in 
the  field;  then  reset  on  Polaris,  lower  the  telescope,  and  observe  the  instant 
of  transit.  The  intervals  which  it  is  necessary  to  wait  before  Polaris  is 
in  the  meridian  are  given  in  the  following  table  for  the  two  stars  men- 

TABLE  A, 


Interval  be- 

Interval be- 

Date. 

tween  6  Cassi- 
opeiae and 

tween  f  Ursae 
Majoris  and 

Polaris. 

Polaris. 

1910 

7m-3 

6m.  5 

1920 

ii    -5 

10    .3 

tioned  and  for  the  years  1910  and  1920.  These  intervals  are  nearly 
correct  when  Polaris  is  either  above  or  below  the  pole.  In  high  latitudes 
it  will  in  general  be  necessary  to  use  the  star  which  is  at  lower  culmination 
at  the  time  of  the  observation.  In  low  latitudes  it  may  be  more  conven- 
ient to  use  the  star  which  is  at  upper  culmination. 


26 


AZIMUTH 


The  precision  with  which  the  time  must  be  observed,  when  using  this 
method,  may  be  judged  from  the  fact  that  at  the  instant  of  culmination 
the  azimuth  of  Polaris  is  changing  at  the  rate  of  about  i '  of  angle  in  2m 
of  time  (in  the  latitudes  of  the  United  States).  Hence  it  will  be  seen 
that  although  Polaris  is  in  the  most  unfavorable  position  for  a  precise 
azimuth  observation,  yet  even  at  this  time  its  azimuth  may  readily  be 
obtained  within  i'  of  the  true  value. 

26.  Azimuth  of  Polaris  when  the  Time  is  Known.  —  Measure  the  angle 
between  Polaris  and  the  mark,  noting  the  time  of  pointing  on  the  star. 
If  possible,  measure  the  altitude  of  the  star. 

(a)  Express  the  watch  time  as  local  time  by  first  applying  any  known 
error  of  the  watch  and  then  adding  4m  for  each  degree  that  the  place  is 
east  of  the  standard  meridian;  or  subtract  4m  for  each  degree  if  the 
place  is  west.     If  the  time  is  A.M.,  add  i2h  and  subtract  one  day  from 
the  date. 

(b)  Interpolate  in  the  following  table  for  the  time  of  upper  culmina- 
tion for  the  date  and  year  as  directed  below. 

TIME  OF  UPPER  CULMINATION  OF  POLARIS  FOR  1915. 


Jan.     i 
Jan.   15 
Feb.     i 

6»    46?9 
5      Si  -6 
4      44-5 

July     i 
July  15 
Aug.    i 

i8h     si°i 
17       56.3 
16       49-7 

Proportional  Part  for  Days. 

d                         m 

Feb.  15 

3       49-2 

Aug.  15 

IS       SS-o 

I                   3-9 

2                           7.8 

Mar.    i 

2      54-o 

Sept.    i 

14       48.4 

3                 n.8 

Mar.  is 

i       58.8 

Sept.  15 

13       53-5 

4                 iS-7 

Apr.     i 

o      Si-9 

Oct.     i 

12       50.7 

5                 19-6 

Apr.  15 

23      52.9 

Oct.   15 

ii       55-8 

6                23.5 

7                27.4 

May    i 

22         SO.O 

Nov.    i 

10       48.9 

8                31-4 

May  15 

21         55-1 

Nov.  15 

9       53-8 

9                 35-3 

June    i 

20         48.5 

Dec.     i 

8       50.8 

10                 39.2 

June  15 

19      53-7 

Dec.  15 

7       55-6 

For  1916,  add  ims  before  Mar.  i ;  subtract  i™6  on  and  after  Mar.  I. 
For  1917,  subtract  0^7.    For  1918,  add  o™9.     For  1919,  add  2™s. 
For  1920,  add  4™o  before  Mar.  i;  add  .1  on  and  after  Mar.  i. 

(c)  Take  the  difference  between  the  time  computed  under  (a)  and 
(6) ;  add  ios  for  each  hour  in  this  interval,  and  then  express  the  result 
in  degrees  and  minutes,  remembering  that  ih  =  15°  and  im  =  15'. 

(d)  Enter  Table  VIII  (or  IX)  with  this  number  of  degrees  and  find 
p  sin  t.     If  the  number  of  degrees  exceeds  180,  subtract  180°  and  look 
opposite  the  remainder  for  p  sin  /.     Correct  this  for  altitude  by  Table  IX 
(or  XII),  employing  preferably  the  measured  altitude;  otherwise  use 
the  latitude  of  the  place. 


ACCURATE  DETERMINATION  OF  AZIMUTH  27 

(e)  The  result  is  the  azimuth  of  Polaris.  If  the  time  of  upper  cul- 
mination is  greater  than  the  local  time  of  the  observation,  the  star  is 
east  of  the  meridian,  unless  the  number  of  hours  exceeds  12,  in  which 
case  it  is  west.  If  the  time  of  culmination  is  less  than  the  local  time,  the 
star  is  west  if  the  number  of  hours  is  less  than  12,  otherwise  it  is  east. 

EXAMPLE. 

On  May  8,  1917,  in  lat.  40°  N.,  long.  71°  W.,  the  angle  is  measured  between  Polaris 
and  a  reference  mark,  the  observation  being  taken  at  7h  45™  P.M.,  Eastern  Time;  the 
watch  is  im  fast ;  altitude  of  Polaris  =  39°  06'. 
Observed  time  7h  45?o 

Error  i.o 

7     44.0  May  8        22     22.6        1915 

Longitude  corr.  16.0 

Upper  culm.  22     21.9        1917 

Local  time  8h  oo?o  Local  time 

14  1/3  X  ios 


From  Table  VIII,  using  216°  05'  —  180°  =  36°  05',  for  1917,  we  find  p  sin  /  =  40'. i. 
From  Table  IX,  for  p  sin  t  =  40'.!  and  altitude  39°  05',  the  correction  =  n'.s. 
The  azimuth  of  Polaris  is  therefore  o°  si'.6,  and  is  west  according  to  (e). 

27.   Accurate  Determination  of  Azimuth  by  Observation  on  Polaris.  — 

An  accurate  determination  of  azimuth  may  be  made  by  using  a  method 
similar  to  that  of  Article  19  except  that  the  determination  of  the  hour 
angle  must  be  more  precise,  and  the  angle  between  the  pole-star  and  the 
azimuth  mark  must  be  measured  with  greater  accuracy. 

The  determination  of  the  hour  angle  should  be  made  by  observing 
several  altitudes  in  quick  succession,  with  their  corresponding  watch 
readings,  on  some  star  which  can  be  identified  (called  the  lime-star)  and 
which  is  nearly  east  or  west  at  the  time  of  the  observation.  Following  is 
a  list  of  bright  stars  which  may  be  used  for  time  determinations  in  the 
northern  hemisphere.  The  position  of  these  stars  may  be  found  by 
consulting  the  star  maps  on  pp.  62-3.  The  exact  right  ascension  and 
declination  for  any  date  must  of  course  be  taken  from  the  list  of  Apparent 
Places  given  in  the  Nautical  Almanac.  In  identifying  stars  by  means  of 
the  chart  the  observer  should  be  on  the  lookout  for  the  planets.  These 
are  not  fixed  in  position  and  hence  cannot  be  shown  on  the  chart.  If  a 
very  bright  star  is  seen  which  cannot  be  found  on  the  chart  it  is  a  planet 
and  should  not  be  used  for  time  observations  unless  it  can  be  positively 
identified  and  its  exact  position  for  the  date  determined. 


28 


AZIMUTH 


LIST   OF    FIXED    STARS 


Constellations  and 
Letters. 

Name. 

Right 
Ascension. 

Declination. 

a.  Arietis  

Hamal  

2h  oim 

4-  °^°  02' 

a  Tauri  

Aldebaran  

4.      3O 

ft  Orionis  

Rigel           

5IO 

—   8     18 

a  Orionis  

Betelgeux            .  .  . 

c       CQ 

+     7       23 

a  Can.  Maj  

Sirius  

6      41 

7      *6 

—  TO       3C 

a  Can.  Min  

Procyon  

7      34 

+     c       27 

a  Hydrae  

Alphard.  

0      23 

—   8     ic 

a  Leonis  

Regulus  

IO       O3 

-f-  12       2  C, 

3  Leonis 

Denebola 

1  1       4.4. 

ct  Virginis 

Spica 

I  3       2O 

~  *3       UJ 

ct  Bootis 

Arcturus 

14.       1  1 

a  Cor    Bor 

Alphecca 

I  ^       IO 

-|-  27       OI 

<x  Ophiuchi 

Ras-Alhague 

~  7      3O 

+  12        38 

a  Lyraj 

Vega 

/     y 

18     33 

+  38       42 

ct  Aquilffi 

Altair 

IO      46 

+    8       C7 

a  Cvcni 

Arided    

20       38 

4-  44      c7 

ct  Pegasi  

Markab  

23     oo 

-I-  14       42 

28.  Determining  the  Hour  Angle.  —  If  we  know  the  latitude  of  the 
place,  the  declination  of  the  star,*  and  the  mean  of  the  altitudes  (corrected 
for  refraction)  we  can  compute  the  hour  angle  of  the  star  by  the  formula 

log  vers  /  =  log  [sin  f  90°  -  (Lat.-Decl.)  j  -sin  Alt.] 

+  log  sec  Lat. 

+  log  sec  Decl.     [6] 

This  is  of  the  same  general  form  as  equation  [i]  and  may  be  solved  by 
means  of  Tables  III  to  VI.  The  value  of  t  thus  found  is  the  hour  angle 
of  the  time-star  corresponding  to  the  mean  of  the  observed  times. 

In  order  to  find  the  hour  angle  of  Polaris  we  take  from  the  Nautical 
Almanac  the  right  ascension  of  Polaris  and  the  right  ascension*  of  the 
time-star  for  the  date  of  the  observation;  the  difference  between  these  is 
the  same  as  the  difference  between  the  hour  angles  of  the  two  stars  at  any 
time.  Combining  this  difference  of  hour  angle  with  the  computed  hour 
angle  of  the  time-star  we  obtain  the  hour  angle  of  Polaris  at  the  instant 
T,  the  mean  of  the  watch  readings.  If  the  right  ascension  of  the  time- 
star  is  less  than  that  of  Polaris  the  difference  is  to  be  subtracted  from  the 
hour  angle  of  the  time-star  to  obtain  the  hour  angle  of  Polaris.  If  the 
right  ascension  of  the  time-star  is  greater  than  the  right  ascension  of 
Polaris  the  difference  is  to  be  added  to  the  hour  angle  of  the  time-star  to 
obtain  the  hour  angle  of  Polaris. 

*  Found  in  the  table  of  Fixed  Stars,  Apparent  Places,  Nautical  Almanac. 


COMPUTING  THE  AZIMUTH  29 

Since  errors  of  the  vertical  angle  will  produce  errors  in  the  computed 
hour  angle  it  is  advisable  to  make  observations  on  two  time-stars,  one 
of  which  is  east  of  the  meridian  and  one  west.  The  mean  of  the  two 
results  for  the  hour  angle  of  Polaris  will  be  nearly  free  from  such 
errors. 

29.  Determining  the  Azimuth.  —  We  now  measure,  by  several  repeti- 
tions, the  angle  between  Polaris  and  the  azimuth  mark,  noting  the  time 
at  each  pointing  on  the  star.     For  the  best  results  a  set  of  repetitions 
should  be  made  with  the  telescope  direct  and  another  set  with  the  telescope 
reversed.     Several  such  sets  may  be  made,  if  desired,  to  increase  the 
accuracy.     The  altitude  of  Polaris  should  be  measured  before  and  after 
each  set. 

30.  Computing  the   Azimuth.  —  In  computing  the  azimuth  of  the 
mark  from  the  transit  it  will  be  better  to  reduce  each  half-set  separately. 
In  the  first  half-set  take  the  mean  of  the  observed  watch  times   (on 
Polaris)   and  call  this  Tf,  which  corresponds   (nearly)  to  the  average 
angle  between   the  star  and  the  mark  in  this  half-set.     The  interval 
T'  —  T  is  the  amount  which  the  hour  angle  of  Polaris  has  increased 
since  the  instant   T,  when  the  time  observation  was  made.     Strictly 
speaking  this  interval  is  in  solar  units  of  time  and  should  be  reduced  to 
sidereal  units.     This  reduction  may  be  made  with  sufficient  accuracy  by 
increasing  the  interval  T'  —  T  by  is  for  each  6m  053  of    time  in  the 
interval,  i.e.,  if  i8m  158  elapsed  between  the  two  observations  the  true 
interval  is  18™  15s  +  3s  =  18™  i8».     This  corrected  interval  added   to 
the  hour  angle  of  Polaris  at  the  time  T  gives  the  hour  angle  of  Polaris 
at  the  time  Tf,  when  the  angles  were  measured.     The  azimuth  of  the 
star  at  this  instant  T'  may  be  found  with  sufficient  accuracy  by  the 
formula 

Azimuth  =  p  sin  t  sec  h,  [4] 

in  which 

p  is  the  polar  distance  for  the  date  of  the  observation,  taken  from  the 
Nautical  Almanac. 

/  is  the  hour  angle  of  Polaris,  just  computed,  at  the  time  T',  and 
h  is  the  altitude  of  the  star  at  the  time  Tf,  found  by  interpolating  between 
the  altitudes  measured  just  before  and  just  after  the  angle  measurements. 
The  azimuth  is  +  if  the  star  is  west  of  the  meridian,  —  if  east,  and  is 
reckoned  from  the  north  point  of  the  horizon.  If  the  latitude  of  the 
place  is  known  and  it  is  not  convenient  to  measure  the  altitude  when  the 
observations  are  made,  it  may  be  found  by  the  equation 

Altitude  =  Latitude  +  p  cos  /  —  K,  [7] 

where  p  cos  /  is  the  quantity  given  in  Tables  VIII  and  XI,  and  K  is  a 
correction  found  in  Table  XIII.     The  second  half-set  of  angles  is  to  be 


30  AZIMUTH 

reduced  in  a  similar  manner.  The  azimuth  of  the  star  for  each  half-set 
is  to  be  combined  with  the  mean  angle  obtained  from  the  repetitions,  thus 
giving  the  azimuth  of  the  mark.  The  interval  of  time  during  a  half-set 
should  be  kept  as  short  as  possible  if  the  best  results  are  desired,  because 
the  azimuth  of  the  star  at  the  mean  of  the  observed  times  is  not  strictly 
the  same  as  the  mean  of  the  different  azimuths,  on  account  of  the  curva- 
ture of  the  star's  path. 

The  determination  of  the  hour  angle  may  be  made  by  several  other 
methods,  such  as  observing  the  transit  of  the  time-star  across  the  meri- 
dian,* across  the  vertical  circle  through  Polaris,  or  by  equal  altitudes  of 
two  stars. 

The  following  examples  will  serve  to  illustrate  the  method  of  com- 
puting the  azimuth: 


EXAMPLE  i. 

AZIMUTH  OBSERVATION. 

Approximate  latitude  42°  21'.  Feb.  u,  1908. 

OBSERVATION  FOR  THE  TIME. 
Altitudes  of  Regulus  (east).  Times. 

17°  05'  7h  i2m  16" 

17   3i  14    31 

17  49  16    07 

18  02  17       20 


Mean   17°  3?' 
Refraction  3' 

Alt.     17°  34' 


HORIZONTAL  ANGLES  FROM  AZIMUTH  MARK  TO  POLARIS. 
(Mark  east  of  star.) 

Altitude 
43°  03' 


Tel.  Direct 
Mark               o°  oo' 
3d   rep.  201°  48' 

Mean            67°  16' 
Tel.  Reversed 
Mark               o°  oo' 
3d    rep.  201°  54' 

Times  —  on  Polaris 
7"  20-  38' 
23    oo 
23    56 

7*  27m  09" 
28     17 

29       21 

43°  01' 
Me?n  67°  18'  :T=7h  28">  16* 

*  An  approximate  meridian,  found  by  one  of  the  methods  previously  given,  will 
usually  be  sufficiently  accurate  for  this  time  observation. 


COMPUTING  THE  AZIMUTH  31 

COMPUTATION  OF  THE  HOUR  ANGLE  OF  REGULUS. 


nat.  sin=  .8666 
nat.  sin=  .3018 

Latitude    42°  21' 
Declination  +  1  2°  25' 

Difference     29°  56' 
Co     60°  04' 
Altitude     1  7°  34' 

log  sec.       .1313 
log  sec.       .0102 

log  9-7510 
log  vers=9.8934 

Diff  .=  .5648 

Right  Ascension  Polaris  =    i    25     32.4 
Diff.  Right  Ascension  =  8h  37m  56».7 

=  129°  29'  /     =  -   77°  26'  at  7*  i5m  031 

=     129°  29' 

Hour  Angle  of  Polaris    ) 

at  7h  I5m  03s        }  -     S^  03 


HOUR  ANGLES  OF  POLARIS 
Intervals 

ist  half-set         2nd  half-set       Hour  Angle  of  Polaris     Corresponding  Time 
r=7>>22«»3i»      7h28mi6*  53°  55'  7*  22m  31 

T=7    15     03       7    15    03  55°  22  7   28    16 


T'-T=        7">28' 
Red.  to  ^ 
sidereal  ) 

13»  I3. 

2 

Interval  =   7™  299 

52°  03' 

I3m  I5» 

3°  19' 
52°  03' 

Hour  Angle     53°  55' 

55°  22' 

COMPUTATION  OF  AZIMUTH. 
First  half-set.  Second  half-set. 


log  p  =  i  .8503 
log  sin  t=g.  9075 
log  sec  h  =  .1361 


log  azimuth  =  i  .8939 
azimuth  =     78'  .32 
azimuth  =  i°  i8'-32 
Angle  =67°  1  6' 


Azimuth  of  line,  N.  65°  57'  .7  E.  N.  65°  s8'.3  E. 

Mean  azimuth  of  mark=  245°  58'  JQ 


32  AZIMUTH 

EXAMPLE  2. 

AZIMUTH  OBSERVATION. 

Latitude  42°  03'  Sept.  5,  1906. 

HORIZONTAL  ANGLES  FROM  MARK  TO  POLARIS. 

(Mark  East  of  Star) 

Tel.  Direct  Times  —  on  Polaris. 

Vernier  6h  39™  48* 

o°  oo'  ob"  40     57 

6th  Rep.  211°  45'  30"  41     39 

Mean    35°  17'  35"  42     29 

43  13 

44  01 

Mean  6h  42™  oi8.i=  T 
Tel.  Reversed 

o°  oo'  oo"  6h  50™  15* 

6th  Rep.  211°  32'  oo"  51     16 

Mean    35°  15'  20"  52     52 

54  26 

55  37 

56  55 


T=  7h  52""  26"  (mean) 


TIME  OBSERVATIONS. 

Mean  Altitude  of  Arcturus  =  27°  12'        (West) 
Refraction  =          2' 


Reduced  Altitude  =27°  10' 


COMPUTATION  OF  HOUR  ANGLE  OF  ARCTURUS. 


nat.  sin  =  .9247 
nat.  sin  =  4566 

Latitude  =42°  03' 
Declination  =  19   40.4 

log  sec. 
log  sec. 

.1292 
•0263 

Difference  =22°  2  2'  .6 
Co  =67°  37'  4 
Altitude  =  27°  10' 

Diff.  =  .4681  log         9.6703 

log  vers=9.8258 

Right  Ascension  Arcturus=i4k  nm  221.6  /  =   70°  42'  at  7h  52m  26" 

Right  Ascension  Polaris=   i    26    21.7  191    15 


Diff.  Right  Ascension=  i2h  45m  oo§.9     Hour  Angle 
=  191°  15'  of  Polaris 


=  261°  57'  at 


MERIDIAN  BY  POLARIS  AT  ELONGATION 


33 


HOUR  ANGLES  OF  POLARIS. 


Intervals 
ist  half-set  2nd  half-set 


T'         6h  42m   oi  '.I 

T         7    52     26 

6h  53m  33a-5 

7    52     26 

T—  r=-ih  iom  259 

Red.  to  i 

Sidereal  j 

10 

Interval=  —  ih  iom  37" 
=  -    17°  39' 
261°  57' 

—  oh  59m  02' 

-    14°  46' 
261°  57' 

Hour  Angle  of  Polaris         Time 


244°  1 8' 
247°  «' 


6h  42m  OIs 

6h    53m338-5 


Altitudes 

Latitude  =  42°  03'  &    42°  03'  .o 
P  cos  /=     —31  .1         —27.8 

4i°3i'.g    41°  35'. 2 
K  (Table  XIII)  .5        ,         .5 

Altitude     41°  31'. 4     41°  34'.; 


COMPUTATION  OF  THE  AZIMUTH. 


Direct  Reversed 

p  cos  /  =  —  31'.  13  —27' .84 

log  p  cos  /  =  i  .4932  n.  i .4446  n. 

log  cos  /  =9.6372  n.  9.5886  n. 

log  #=1.8560*  1.8560 

log  sin  /  =9.9548  n.  9.9646  n. 

log  sec  h  =0.1257  0.1261 

log  Azimuth  =  1.9365  n.  1-9467  n. 

Azimuth  =  —        86' .40  —88'  .45 

=  —    i°  26'  24"  —    i°  28'  27" 

Angle         35    17  ,35  35    15  20 

Azimuth  of  line  =     36°  43'  59''  36°  43'  47" 
Mean,  N  36°  43'  53". o  E. 


31.  Meridian  by  Polaris  at  Elongation.  —  If  Polaris  is  at  its  extreme 
east  or  west  elongation,  i.e.,  if  it  has  its  greatest  east  or  west  bearing,  its 
azimuth  can  be  accurately  determined  without  knowing  the  exact  time 
at  which  elongation  occurs.  The  approximate  time,  near  enough  to 
determine  when  the  observation  should  be  begun,  may  be  taken  from 
Table  B.  The  positions  of  the  constellations  at  the  times  of  elongations 
may  be  seen  by  reference  to  Fig.  3. 

*  p  cos  /  is  computed  by  adding  upward,  and  the  azimuth  by  adding  downward. 


34 


AZIMUTH 


TABLE  B.* 
APPROXIMATE    LOCAL  TIMES    OF  ELONGATION   OF  POLARIS. 


Date. 

Eastern 
Elongation. 

Western 
Elongation. 

Date. 

Eastern 
Elongation. 

Western 
Elongation.    * 

h     m 

h     m 

h     m 

h     m 

Jan.     i 

12   50P.M. 

12   40A.M. 

July     I 

12   58  A.M. 

12   48  P.M. 

"     15 

ii  55  A.M. 

II   45  P.M. 

"       15 

II    59P.M. 

ii  53  A.M. 

Feb.    i 

10   48 

10   38 

Aug.    i 

1°  53 

10  47 

"      15 

9  52 

9  42 

"     15 

9  58 

9  S2 

Mar.    i 

(i 

8  57 

8  47 

Sept.   i 

851 

8  45 

15 

8   02 

7  52 

"      i5 

7  57 

7  5i 

Apr.     i 

6  55 

6  45 

Oct.     i 

6  54 

6  48 

15 

6  oo 

5  5° 

"     15 

5  59 

5  53 

May    i 

4  57 

4  47 

Nov.    i 

4  52 

4  46 

"      15 

4  02 

3  52 

"      15 

3  57 

3  Si 

June    i 

2    56 

2  46 

Dec.    i 

2  54 

2    48 

"      15 

2   OI 

i  5i 

"     15 

i  59 

i  53 

TABLE   C. 
POLAR   DISTANCES   OF  POLARIS. 


1911 

0  10'  07".82 

1916 

i°  08'  34".97 

1912 

09  49.22 

1917 

i  08  16.45 

1913 

09  30.64 

1918 

i  07  57.94 

1914 

09  12.07 

1919 

i  07  39.45 

1915 

08  53-51 

1920 

i  07  20.98 

In  order  to  make  this  observation  set  the  transit  in  position  a  half  hour 
or  more  before  the  pole-star  reaches  its  eastern  or  western  elongation. 
Set  the  vertical  cross-hair  on  Polaris  and  follow  it,  with  the  plate  tangent 
screw,  as  long  as  the  star  continues  to  move  away  from  the  meridian. 
When  the  star  is  near  its  greatest  elongation  it  will  appear  to  move  verti- 
cally for  a  few  minutes  and  then  will  begin  to  move  back  toward  the 
meridian.  While  the  star  is  in  this  extreme  position  it  should  be  carefully 

*  To  find  the  Local  Time  for  any  other  date  than  the  first  or  isth,  interpolate  between 
the  values  given  in  the  table;  the  daily  change  is  about  4  minutes.  To  convert  this 
Local  Time  into  Standard  Time  take  the  diference  between  the  longitude  of  the  place 
and  the  longitude  of  the  Standard  meridian  (expressed  in  hours,  minutes,  and  seconds) 
and  add  this  difference  to  the  Local  Time  if  the  place  is  west  of  the  Standard  meridian, 
subtract  if  the  place  is  east. 


MERIDIAN  BY  EQUAL  ALTITUDES  OF  A  STAR         35 

bisected  with  the  vertical  cross-hair  and  a  stake  set  at  some  convenient 
distance  from  the  instrument  in  line  with  the  cross-hair.  In  order  to 
eliminate  the  errors  due  to  poor  adjustment  of  the  transit  the  telescope 
should  be  immediately  reversed,  the  star  again  bisected  and  another 
point  set  in  line  with  the  vertical  cross-hair.  The  mean  of  these  two 
points  will  give  the  direction  of  the  star  at  elongation.  The  angle 
between  this  direction  and  the  direction  of  the  meridian  may  be  cal- 
culated by  the  formula 

sin  azimuth  =  sin  polar  distance  X  sec  latitude,  [8] 

or,  with  sufficient  accuracy, 

azimuth  (in  seconds)  =  polar  distance  (in  seconds)  X  sec  latitude.     [9] 

This  calculated  azimuth  may  be  laid  off  either  by  means  of  the  transit, 
using  repetitions,  or  by  measuring  the  distance  from  the  transit  to  the 
point  that  was  set,  and  then  calculating  the  perpendicular  offset  which 
will  give  a  point  exactly  north  of  the  transit.  This  offset  should  be  laid 
off  with  a  steel  tape.  If  desired,  angles  to  some  mark  may  be  measured 
instead  of  setting  stakes  in  line  with  the  cross-hair.  At  elongation 
Polaris  changes  its  azimuth  so  slowly  that  there  will  usually  be  time  to 
set  several  points  or  to  measure  several  angles  before  the  star  changes  its 
bearing  as  much  as  5  seconds  of  angle.  In  latitude  40°  N.,  for  example, 
the  azimuth  of  Polaris  30  minutes  before  or  after  elongation  is  about  one 
minute  (of  angle)  less  than  it  is  at  elongation.  The  change  in  azimuth 
varies  as  the  square  of  the  time  interval,  hence  in  iom  either  side  of 
elongation  the  azimuth  would  vary  about  6  to  7  seconds. 

The  polar  distance  of  Polaris  may  be  taken  from  Table  C,  or  from 
Table  VIII  under  p  sin  /  and  opposite  H.  A.  =  90°.  If  the  latitude 
is  not  known,  the  observed  altitude  of  Polaris  may  be  taken  as  approxi- 
mately equal  to  it.  The  error  resulting  from  this  assumption  will  be 
about  i"  of  azimuth  for  each  i'  error  in  the  latitude. 

32.  Meridian  by  Equal  Altitudes  of  a  Star.  —  A  very  simple  method  of 
determining  the  direction  of  the  meridian  is  by  observing  on  a  star  at 
equal  altitudes  on  opposite  sides  of  the  meridian.  This  method  is  accu- 
rate and  requires  no  Nautical  Almanac  or  tables  of  any  kind ;  it  is  not 
convenient,  however,  as  it  requires  two  observations  at  night  separated 
by  several  hours'  time,  but  it  may  prove  of  value  when  for  some  reason 
the  more  rapid  and  convenient  methods  are  not  available,  as,  for  instance, 
in  the  southern  hemisphere  where  there  is  no  bright  star  near  the  pole. 

In  order  to  determine  the  true  meridian  select  some  star  which  is  not 
far  from  the  pole  and  which  is  on  the  west  side  of  the  meridian  in  about 
the  position  of  A,  Fig.  6.  The  hour  angle  should  be  such  that  the  star 


36  AZIMUTH 

will  reach  an  equal  altitude  on  the  east  side  of  the  meridian  about  6h  or 
8h  later,  so  that  the  second  half  of  the  observation  will  occur  before 
daylight.  One  of  the  stars  in  Cassiopeia  could  be  used,  for  example,  the 
first  observation  being  made  when  the  star  has  an  hour  angle  of  about 
135  degrees  and  the  second  at  an  hour  angle  of  225  degrees,  as  shown  at 
A  and  A'  in  Fig.  6.  The  star  is  bisected  with  both  cross-hairs,  the 
horizontal  circle  is  clamped,  and  the  altitude  is  read  and  recorded.  The 
telescope  is  then  lowered  and  a  point  set  in  line.  Some  memorandum 
or  sketch  should  be  made  for  identifying  the  star  at  the  second  observa- 
tion. When  the  star  is  approaching  the  same  altitude  on  the  east  side  of 


HORIZON 

Evening-  Observations.  Morning  Observations. 

FIG.  6.    Meridian  by  Equal  Altitudes  of  a  Circumpolar  (Northern  Hemisphere). 

the  meridian  the  telescope  is  set  at  exactly  the  same  altitude  as  was  read  at 
the  first  observation.  The  star  is  then  bisected  with  the  vertical  cross- 
hair, and  followed  until  it  passes  the  horizontal  cross-hair.  After  this 
instant  the  tangent  screw  should  not  be  touched.  Another  point  is  then 
set  in  line  with  the  vertical  cross-hair  as  before.  The  bisector  of  the 
angle  between  these  two  points  is  the  meridian  line  through  the  instru- 
ment. If  desired,  several  pairs  of  altitudes  may  be  observed,  to  increase 
the  accuracy,  as  h  and  h,  h'  and  h' ',  Fig.  6.  Each  pair  should  be  combined 
independently  of  the  others.  If  preferred,  angles  may  be  measured 
from  some  reference  mark  instead  of  setting  points  in  line  with  the  star. 
Instead  of  taking  the  altitudes  at  random  it  is  well  to  set  the  telescope  so 
that  the  vernier  reads  some  whole  degree  or  half  degree  and  then  set  the 
vertical  cross-hair  on  the  star  and  follow  it  with  the  upper  tangent  screw 


MERIDIAN  BY  EQUAL  ALTITUDES  OF  THE  SUN       37 

until  both  cross-hairs  bisect  the  star.  If  the  transit  has  a  full  vertical 
circle,  errors  in  the  instrument  may  be  eliminated  by  taking  the  evening 
observations  with  the  instrument  direct  and  the  morning  observations 
with  the  instrument  reversed. 

33.  Meridian  by  Equal  Altitudes  of  the  Sun.  —  The  observation  just 
described  may  also  be  made  on  the  sun  at  equal  altitudes  in  the  forenoon 
and  afternoon,  the  difference  being  that  the  sun  changes  its  declination 
during  the  interval  between  the  observations  and  hence  a  correction  must 
be  applied  in  order  to  obtain  the  direction  of  the  true  meridian.  Since 
the  change  for  a  given  date  is  practically  the  same  each  year  no  Nautical 
Almanac  is  necessary,  but  the  hourly  change  in  declination  for  any  date 
may  be  taken  from  Table  XV. 

The  observation  is  made  as  follows:  At  some  time  in  the  forenoon,  say 
between  8h  and  ioh  A.M.,  set  up  the  transit  and  set  the  plate  vernier  at 
o  degrees.  Point  the  telescope  at  some  azimuth  mark  and  clamp  the 
lower  plate.  Loosen  the  upper  plate  and  turn  to  the  sun.  Set  the 
vertical  arc  so  that  the  vernier  reads  some  whole  degree  or  some  even 
10  minutes  (higher  than  the  sun);  then  set  the  vertical  cross-hair  on  the 
left  edge  of  the  sun  and  follow,  with  the  upper  tangent  screw,  until  the 
lower  edge  of  the  sun  is  in  contact  with  the  horizontal  cross-hair.  Note 
the  time  and  read  the  altitude  and  the  horizontal  angle.  In  the  after- 
noon, a  little  before  the  sun  reaches  this  same  altitude,  set  the  vernier  on 
o  degrees  and  point  again  on  the  mark.  Set  the  telescope  so  that  the 
vernier  of  the  vertical  arc  reads  the  same  altitude  as  was  used  for  the 
A.M.  observation.  When  the  sun  comes  into  the  field  of  the  telescope  set 
the  vertical  cross-hair  on  the  right  edge  of  the  sun  and  follow  it,  with  the 
upper  plate  tangent  screw,  until  the  lower  edge  of  the  sun  is  again  in 
contact  with  the  horizontal  cross-hair.  Note  the  time  and  read  the 
horizontal  angle. 

The  mean  of  the  two  readings  of  the  horizontal  angle  is  approximately 
the  angle  between  the  mark  and  the  south  point.  Since,  however,  the 
sun's  declination  is  not  the  same  at  the  two  observations  it  will  be  neces- 
sary to  apply  the  correction 

2  cos  Lat.  X  sin  Hour  Angle 

in  which  d  is  the  hourly  change  from  Table  XV  multiplied  by  the  num- 
ber of  hours  between  the  observations,  and  the  Hour  Angle  equals  half 
this  number  of  hours  turned  into  degrees.  In  the  table,  the  +  sign  indi- 
cates that  the  sun  is  going  north,  the  —  sign  indicates  that  it  is  going 
south.  If  the  sun  is  going  north  the  mean  of  the  two  angles  gives  a  point 
west  of  the  true  south;  if  the  sun  is  going  south  the  mean  angle  is  to  a 
point  east  of  south. 


AZIMUTH 


EXAMPLE. 

Latitude  42°  18'  N. 
A.M.  Observation 
Reading  on  mark  o°  oo' 
Pointings  on  left  and  upper  edges 
Altitude  24°  58' 

Horizontal  Angle*  357°  14'  15" 
Time  7h  19™  30*  AM. 

%  elapsed  time=  4h  26™  221 
'=66°  35'  30" 

log  sin  ^=9.9627 
log  cos  L  =9.8690 

9-8317 

log  230"  .9=  2. 3634 

2-5317 

Correction=  -  340" .2=  -  5*  40" .2 


April  19,  1906. 
P.M.  Observation 
Reading  on  mark  o°  oo' 
Pointings  on  right  and  upper  edges 
Altitude  24°  58' 
Horizontal  Angle*  162°  28'  oo" 
Time  4h  12™  15"  P.M. 

Change  in  declination  in 


=  2  30"  .9 

Mean  Plate  Reading  259°  51'  08" 
1  80° 


Uncorrected  bearing    79°  51'  08" 
Correction         —  5'  40" 

Corrected  bearing  S.    79°  45'  28"  E. 
Azimuth=  280°  14'  32" 


*  Read  "clockwise"  in  each  case. 


TABLES 

I.  REFRACTION  CORRECTION. 
II.  CONVERGENCE  OF  MERIDIANS. 

III.  LOGARITHMIC  SECANTS. 

IV.  NATURAL  SINES. 

V.  LOGARITHMS  OF  NUMBERS. 
VI.   LOGARITHMIC  VERSED  SINES. 
VII.   HOUR  ANGLES  OF  d  CASSIOPEIA. 
VIII.   COORDINATES  OF  POLARIS. 
IX.   CORRECTION  FOR  AZIMUTH. 
X.   HOUR  ANGLES  OF  d  DRACONIS. 
XI.   COORDINATES  OF  POLARIS. 
XII.   CORRECTION  FOR  AZIMUTH. 
XIII.  VALUES  OF  K  FOR  COMPUTING  THE  ALTITUDE. 

XIV.  CORRECTION  TO  SUN'S  DECLINATION. 

XV.  SUN'S  DECLINATION. 


AZIMUTH 


TABLE    I.       REFRACTION    CORRECTION. 

(In   Minutes.) 
(True  Alt.  =  Meas'd  Alt.  -  Refr.) 


Alt. 

Refr. 

Alt. 

Refr. 

Alt. 

Refr. 

5° 

9'-  9 

13° 

4'.  i 

25° 

'.i 

6 

8-5 

14 

3-8 

30 

•  7 

7 

7-4 

15 

3-6 

35 

•4 

8 

6.6 

16 

3-3 

40 

.2 

9 

5-9 

17 

3-1 

45 

.0 

10 

5-3 

18 

3-0 

50 

0.8 

ii 

4-9 

19 

2.8 

55 

0.7 

12 

4-5 

20 

2.6 

60 

0.6 

TABLE    II.       CONVERGENCE    OF    THE    MERIDIANS. 
(In  Minutes.) 


Miles  (east  or  west). 

Lat. 

i 

2 

3 

4 

5 

6 

7 

8 

9 

10 

30° 
35 

°'-5 
0.6 

I'.Q 

1.2 

i'.S 
1.8 

2'.0 

2.4 

2'.  5 
3-o 

3'-o 
3-6 

3'-  5 

4-2 

4'.o 
4-9 

4'-  5 

5-5 

S'-O 

6.! 

40 
45 
50 

0.7 
0.9 

I.O 

i-5 
i-7 

2.1 

2.2 
2.6 
3-1 

2.9 

3-5 
4.1 

3-6 
4-3 
5-2 

4-4 

11 

5-i 
6.1 

7-2 

5-8 
6.9 
8-3 

6-5 
7-8 
9-3 

w 

10.3 

TABLES 


TABLE    III.— LOGARITHMIC    SECANTS. 


tf 

Minutes. 

Proportional  Parts. 

Q 

o' 

10' 

2O' 

30' 

40' 

50' 

i' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

o 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

0 

O 

o 

o 

o 

o 

o 

0 

o 

i 

.0001 

.0001 

.0001 

.0001 

.0002 

.0002 

0 

0 

0 

0 

0 

0 

0 

o 

o 

2 

.0003 

i-w-y> 

.0003 

.000^ 

.0004 
.ooo£ 

.0005 

.0005 

0 

o 

o 

o 

o 

o 

o 

0 

I 

3 
4 

.0000 
.0011 

.0011 
__.To 

.0012 

.0013 

.001^ 

.0015 

0 

o 

o 

o 

I 

I 

I 

I 

I 

6 

.0017 

.0024 

.  OOIo 

.0025 

.OOI( 

.0027 

.0028 

.0029 

.0031 

0 

o 

I 

1 

I 

I 

I 

I 

7 
g 

.0032 

.0034 

.0036 

.0037 

.0039 

.0041 

0 

0 

I 

1 

1 

I 

I 

I 

9 

.0054 

.0056 

.0058 

!oo6o 

!oo62 

.0052 

.0064 

0 

I 

I 

I 

I 

I 

2 

2 

IO 

.0066 

.0069 

.0071 

.0073 

.0076 

.0078 

0 

I 

I 

1 

I 

2 

2 

2 

ii 

.0081 

.0083 

.0086 

.0088 

.0091 

.0093 

o 

I 

I 

I 

2 

2 

2 

2 

12 

.0096 

.0099 

.OIOI 

.0104 

.0107 

.0110 

o 

I 

I 

1 

2 

2 

2 

3 

13 

.0113 

.0116 

.0119 

.0122 

.0125 

.0128 

o 

I 

I 

2 

2 

2 

2 

3 

14 

.0131 

.0134 

•  0137 

.0141 

.0144 

.0147 

0 

I 

I 

2 

2 

2 

3 

3 

IS 

.0151 

.0154 

.0157 

.0161 

.016^: 

.0168 

0 

I 

I 

2 

2 

2 

3 

3 

16 

.0172 

.0175 

.0179 

.0183 

.0186 

.0190 

0 

I 

I 

2 

2 

3 

3 

3 

i7 

.0194 

.0198 

.0202 

.0206 

.0210 

.0214 

o 

I 

2 

2 

2 

3 

3 

4 

18 

.021? 

.0222 

.O226 

.0230 

.0235 

.0239 

0 

I 

2 

2 

3 

3 

3 

4 

iQ 

.0243 

.0248 

.0252 

.0257 

.0261 

.0266 

o 

I 

2 

2 

3 

3 

4 

4 

20 

.027O 

.0275 

.O279 

.0284 

.0289 

.0294 

0 

I 

2 

2 

3 

3 

4 

4 

21 

.0298 

•0303 

.0308 

•  0313 

.0318 

•0323 

I 

'2 

2 

3 

3 

4 

4 

5 

22 

.0328 

'°36s 

•0339 

•  0344 
0,76 

.0349 

.0382 

•  0354 

.0387 

I 

I 

2 

2 

2 

3 

3 

4 

4 

5 

23 
24 

.0300 

.0393 

•  0398 

.O37I 
.0404 

•*JO/V-' 

.0410 

.0416 

OyfCT 

.0421 

rMC*7 

I 

7 

2 

2 

3 

3 

4 

5 

5 

25 
26 

.0427 
.0463 

•°433 
.0470 

•  ^43? 

.0476 

.0445 

.0482 

•  U4v>x 

.0488 

•U4o/ 
•0495 

I 

2 

3 

3 

4 

4 

5 

6 

27 

.0501 

.0508 

.0514 

.0521 

.0527 

•0534 

I 

2 

3 

3 

4 

4 

5 

6 

28 

.0541 

•0547 

.0554 

.0561 

.0568 

•0575 

7 

2 

3 

3 

4 

5 

6 

29 

.0582 

.0589 

.0596 

.0603 

.0610 

.0617 

I 

2 

3 

4 

4 

5 

6 

6 

3° 

.0625 

.0632 

•0639 

.0647 

.0654 

.0662 

I 

2 

3 

4 

5 

5 

6 

7 

31 

.0669 

.0677 

.0685 

.0692 

.0700 

.0708 

I 

2 

3 

4 

5 

5 

6 

7 

32 

.0716 

.0724 

.0732 

.0740 

.0748 

.0756 

1 

2 

3 

4 

5 

6 

6 

7 

33 

.0764 

.0772 

.0781 

.0789 

.0797 

.0806 

I 

2 

3 

4 

5 

6 

7 

7 

34 

.0814 

.0823 

.0831 

.0840 

.0849 

.0858 

I 

3 

3 

4 

5 

6 

7 

8 

35 

.0866 

.0875 

.0884 

.0893 

.0902 

.0911 

I 

3 

4 

5 

5 

6 

7 

8 

36 

.0920 

.0930 

•0939 

.0948 

.0958 

.0967 

1 

3 

4 

5 

6 

7 

7 

8 

37 

.0977 

.0986 

.0996 

.1005 

.1015 

.1025 

T 

3 

4 

$ 

6 

7 

8 

9 

38 

•  1035 

.1045 

•  1055 

.1065 

.1075 

.1085 

I 

3 

4 

S 

6 

7 

8 

9 

39 

.1095 

.1105 

.me 

.1126 

.1136 

.1147 

I 

3 

4 

5 

6 

7 

8 

9 

40 

."57 

.1168 

.1179 

.1190 

.1200 

.1211 

T 

3 

4 

5 

6 

7 

9 

0 

4i 

.1222 

•1233 

.1244 

•  1255 

.1267 

.1278 

I 

3 

4 

6 

7 

8 

9 

0 

42 

.1289 

.1301 

.1312 

.1324 

.1335 

•1347 

I 

3 

5 

6 

7 

n 

9 

0 

43 

•1359 

•I37I 

.1382 

.1394 

.1406 

.I4l8 

I 

4 

5 

6 

7 

8 

o 

I 

44 

•1431 

•1443 

.1455 

.1468 

.1480 

•1493 

I 

4 

5 

6 

7 

9 

0 

I 

45 

•1505 

.1518 

.1531 

.1543 

.1556 

.1569 

I 

3 

4 

5 

6 

8 

9 

7O 

2 

46 

.1582 

.•1595 

.1609 

.1622 

•  1635 

.1649 

I 

3 

4 

5 

7 

8 

9 

I 

2 

47 

.1662 

.1676 

.1689 

.1703 

.1717 

.1731 

I 

3 

4 

6 

7 

8 

0 

II 

2 

48 

•1745 

•1759 

•1773 

.1787 

.1802 

.l8l6 

I 

3 

4 

6 

7 

9 

o 

II 

3 

49 

.1831 

.1845 

.1860 

.1875 

.1889 

.1904 

I 

3 

4 

6 

7 

9 

10 

2 

3 

50 

.1919 

•1934 

.1950 

.1965 

.1980 

.1996 

2 

3 

5 

6 

8 

9 

II 

12 

4 

Si 

.2011 

.2027 

•2043 

.2059 

.2074 

.2090 

a 

3 

5 

6 

8 

0 

II 

13 

4 

52 

.2107 

.2123 

.2139 

.2156 

.2172 

.2189 

2 

3 

5 

7 

8 

0 

12 

13 

5 

53 

.220-; 

.2222 

.2239 

.2256 

.2273 

.229O 

2 

3 

5 

7 

Q 

o 

12 

14 

5 

54 

.2308 

•2325 

•2343 

.2360 

.2378 

.  2396 

2 

4 

5 

7 

9 

II 

12 

14 

6 

55 

.2414 

•2432 

.2450 

.2469 

.2487 

.2^06 

2 

4 

6 

7 

9 

I 

13 

15 

7 

56 

.2524 

•2543 

.2^62 

.2581 

.2600 

.2620 

2 

4 

6 

8 

o 

II 

13 

15 

7 

57 

•  2639 

.2658 

.2678 

.2698 

.2718 

•  2738 

2 

4 

6 

8 

0 

12 

14 

76 

8 

58 

.27=^8 

.2778 

.2799 

.2819 

.2840 

.2861 

2 

4 

6 

8 

TO 

T2 

M 

76 

9 

59 

.2882 

.2903 

.2924 

.2945 

•  2967 

.2988 

2 

4 

6 

0 

I 

13 

15 

17 

9 

60 

.3010 

•  3032 

•  3054 

•  3077 

.3099 

..?I22 

2 

a 

7 

n 

T 

3 

76 

18 

20 

AZIMUTH 


TABLE    IV. —  NATURAL    SINES. 


Minutes. 

Proportional  parts. 

& 

Q 

o' 

10' 

20' 

30' 

40' 

50' 

i' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

0 

.0000 

.0029 

.0058 

.0087 

.0116 

.0145 

3 

6 

0 

12 

15 

20 

23 

26 

i 

•  0175 

.0204 

•  0233 

.0262 

.0291 

.0320 

3 

ft 

9 

12 

15 

j 

20 

23 

26 

2 

•  0349 

.0378 

.0407 

•  0436 

.0465 

.0494 

3 

ft 

9 

12 

15 

~ 

20 

2; 

26 

3 

.0523 

•  0552 

.0581 

.0610 

.0640 

.0669 

3 

ft 

9 

12 

15 

7 

20 

23 

26 

4 

.0698 

.0727 

.0756 

.0785 

.0814 

•  0843 

3 

ft 

9 

12 

15 

7 

20 

2  „•> 

26 

5 

.0872 

.0901 

.0929 

•  0958 

.0987 

.1016 

3 

6 

9 

12 

I5 

7 

20 

23 

26 

6 

.1045 

.1074 

.1103 

•  II32 

.  1161 

.1190 

3 

ft 

9 

12 

14 

j 

20 

2,3 

26 

7 

.1219 

.1248 

.1276 

•I30S 

•  1334 

•1363 

3 

6 

9 

12 

i_ 

7 

20 

2c 

26 

8 

•  1392 

.1421 

.1449 

.1478 

•  1507 

.1536 

3 

6 

9 

I  I 

i- 

- 

20 

23 

26 

9 

•  1564 

•  1593 

.1622 

.1650 

.1679 

.1708 

3 

6 

9 

11 

14 

7 

20 

23 

26 

10 

.1736 

•1765 

•  1794 

.1822 

.1851 

.1880 

3 

ft 

9 

I  I 

i_ 

7 

2O 

23 

26 

ii 

.I9o8 

•1937 

.1965 

.1994 

.2022 

.2051 

3 

6 

9 

1  I 

14 

- 

20 

23 

26 

12 

.2079 

.2108 

.2136 

.2164 

•2193 

.2221 

3 

6 

9 

I  I 

i_ 

-. 

2O 

23 

26 

13 

.2250 

.2278 

.2306 

•2334 

•2363 

.2391 

3 

6 

8 

11 

14 

j 

20 

23 

25 

14 

.2419 

.2447 

•  2476 

.2504 

•2532 

.2560 

3 

6 

8 

11 

14 

- 

20 

22 

25 

IS 

.2588 

.2616 

.2644 

.2672 

.2700 

.2728 

3 

6 

8 

II 

14 

- 

20 

22 

25 

16 

.2756 

.2784 

.2812 

.2840 

.2868 

.2896 

3 

6 

8 

11 

14 

7 

20 

22 

25 

i7 

.2924 

.2952 

.2079 

.3007 

•3035 

.3062 

3 

6 

8 

1  1 

14 

20 

aa 

25 

18 

.3090 

.3118 

•  3145 

•3173 

.3201 

.3228 

3 

6 

8 

11 

'4 

7 

20 

22 

25 

19 

-3256 

•3283 

•33" 

•3338 

•3365 

•3393 

3 

5 

8 

II 

14 

16 

«P 

22 

25 

20 

.3420 

.3448 

•3475 

•3502 

•3529 

•3557 

3 

S 

8 

II 

M 

1  6 

19 

22 

24 

21 

•3584 

.3611 

•3638 

•3665 

.3692 

•3719 

3 

5 

8 

II 

14 

1  6 

19 

23 

24 

22 

•3746 

•3773 

.3800 

.3827 

•3854 

•  3881 

3 

5 

8 

II 

12 

16 

19 

21 

24 

23 

•  3907 

•3934 

.3961 

•3987 

.4014 

.4041 

3 

5 

8 

II 

13 

16 

19 

21 

24 

24 

.4067 

.4094 

.4120 

.4147 

•4173 

.4200 

3 

5 

8 

II 

33 

16 

18 

21 

24 

25 

.4226 

•4253 

•  4279 

•4305 

•4331 

•4358 

3 

5 

8 

10 

13 

16 

18 

21 

24 

26 

•  4384 

.4410 

•  4436 

.4462 

.4488 

•4514 

3 

5 

8 

10 

13 

16 

18 

21 

23 

27 

•  4540 

-4566 

•4592 

.4617 

.4643 

.4669 

3 

5 

8 

IO 

13 

15 

18 

21 

23 

28 

•  4695 

•4720 

.4746 

•4772 

•4797 

.4823 

3 

5 

8 

10 

T  ^ 

15 

18 

20 

23 

29 

.4848 

•4874 

.4899 

.4924 

•4950 

•4975 

3 

5 

8 

10 

13 

15 

18 

20 

23 

30 

.5000 

•5025 

•  5050 

•5075 

.5100 

•5125 

3 

S 

8 

TO 

13 

15 

18 

20 

23 

31 

•  5150 

•5175 

.5200 

•5225 

•  5250 

•5275 

5 

7 

10 

12 

i5 

*7 

20 

22 

32 

•  5299 

•5324 

•5348 

•5373 

•5398 

•  5422 

5 

7 

10 

12 

15 

17 

20 

22 

33 

.5446 

•5471 

•5495 

•5519 

•5544 

•5568 

5' 

7 

10 

12 

15 

*7 

19 

22 

34 

•5592 

.5616 

•  5640 

•  5664 

.5688 

•5712 

5.' 

7 

10 

12 

14 

17 

19 

22 

35 

•5736 

.5760 

.5783 

.5807 

•  5831 

.5854 

•-$• 

7 

Q 

12 

14 

17 

19 

21 

36 

•5878 

.5901 

•5925 

•5948 

•5972 

•5995 

S 

7 

9 

12 

14 

16 

19 

21 

37 

.6018 

.6041 

.6065 

.6088 

.6111 

.6134 

5 

7 

9 

12 

14 

16 

18 

21 

38 

•6157 

.6180 

.6202 

.6225 

.6248 

.6271 

5 

7 

9 

II 

14 

16 

18 

20 

39 

.6293 

-6316 

•6338 

•  636! 

•6383 

.6406 

4 

7 

9 

II 

13 

16 

18 

20 

40 

.6428 

•  6450 

.6472 

.6494 

•  6517 

•6539 

4 

7 

9 

I  I 

*3 

15 

18 

20 

4i 

.6561 

•6583 

.6604 

.6626 

.6648 

.6670 

4 

7 

9 

I  I 

13 

15 

17 

2O 

42 

.6691 

•6713 

•6734 

.6756 

.6777 

.6799 

4 

6 

9 

II 

13 

15 

i7 

19 

43 

.6820 

.6841 

.6862 

.6884 

.6905 

.6926 

4 

6 

8 

II 

13 

15 

17 

19 

44 

.6947 

.6967 

.6988 

.7009 

.7030 

.7050 

4 

6 

8 

10 

12 

14 

i? 

19 

i 

o' 

10' 

20' 

30' 

40' 

So' 

i' 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

Minutes. 

Proportional  parts. 

TABLES 

TABLE    IV.  —  (Continued) . 


43 


Minutes. 

Proportional  parts. 

bib 
Q 

o' 

10' 

20' 

30' 

40' 

50' 

i' 

a' 

3' 

4' 

5' 

6' 

f 

8' 

9' 

45 

.7071 

.7092 

.7112 

•7133 

•  7i53 

•  7i73 

4 

6 

8 

JO 

la 

M 

16 

18 

46 

.7193 

.7214 

•7234 

•7254 

.7274 

•  7294 

4 

6 

8 

JO 

12 

*4 

JO 

18 

47 

•  7314 

•7333 

•7353 

•7373 

•7392 

.7412 

4 

6 

8 

JO 

la 

14 

1  6 

18 

48 

•  7431 

•7451 

.7470 

.7490 

•7509 

.7528 

4 

6 

8 

10 

12 

14 

IS 

i7 

49 

•7547 

.7566 

•7585 

.7604 

.7623 

.7642 

4 

6 

8 

0 

II 

13 

IS 

i? 

SO 

.7660 

.7679 

.7698 

.7716 

•7735 

•7753 

4 

6 

7 

9 

II 

13 

15 

17 

51 

.7771 

.7790 

.7808 

.7826 

.7844 

.7862 

4 

5 

7 

0 

II 

13 

14 

16 

52 

.7880 

.7898 

.7916 

•7934 

•7951 

.7969 

4 

s 

7 

0 

1J 

12 

14 

16 

53 

.7986 

.8004 

.8021 

•  8039 

.8056 

.8073 

3 

s 

7 

9 

JO 

12 

14 

IS 

54 

.8090 

.8107 

.8124 

.8141 

.8158 

•8i75 

3 

S 

7 

8 

10 

12 

14 

15 

55 

.8192 

.8208 

.8225 

.8241 

.8258 

.8274 

3 

s 

7 

8 

10 

T2 

13 

15 

56 

.8290 

.8307 

•8323 

•8339 

•8355 

•8371 

3 

5 

6 

8 

IO 

II 

*3 

14 

57 

.8387 

•  8403 

.8418 

•8434 

•  8450 

•  8465 

3 

5 

6 

8 

9 

11 

12 

14 

58 

.8480 

.8496 

•  8511 

.8526 

.8542 

•8557 

3 

5 

6 

8 

9 

n 

ia 

14 

59 

.8572 

•8587 

.8601 

.8616 

•  8631 

.8646 

4 

6 

7 

9 

10 

la 

13 

60 

.8660 

.8675 

.8689 

.8704 

.8718 

.8732 

3 

4 

6 

7 

9 

TO 

12 

13 

61 

.8746 

.8760 

.8774 

.8788 

.8802 

.8816 

3 

4 

6 

7 

8 

10 

II 

13 

62 

.8829 

•  8843 

.8857 

.8870 

.8884 

.8897 

3 

4 

5 

7 

8 

0 

II 

12 

63 

.8910 

.8923 

•  8936 

.8949 

.8962 

•8975 

3 

4 

5 

6 

8 

9 

IO 

13 

64 

.8988 

.9001 

.9013 

.9026 

.9038 

.9051 

3 

4 

5 

6 

8 

9 

10 

II 

65 

.9063 

•9075 

.9088 

.9100 

.9112 

.9124 

4 

5 

6 

'  7 

S 

10 

II 

66 

•  9135 

.9147 

•9159 

.9171 

.9182 

.9194 

3 

5 

6 

7 

'  8 

.  9 

10 

67 

.9205 

.9216 

.9228 

•9239 

.9250 

.9261 

3 

4 

6 

7 

8 

9 

IO 

68 

.9272 

.9283 

•9293 

•9304 

•  9315 

•9325 

3 

4 

S 

6 

8 

9 

10 

69 

•9336 

•9346 

•9356 

•9367 

•  9377 

•9387 

3 

4 

5 

6 

7 

8 

9 

70 

•  9397 

.9407 

.9417 

.9426 

•9436 

.9446 

3 

4 

5 

6 

7 

8 

9 

71 

•9455 

•9465 

•  9474 

•9483 

•9492 

.9502 

3 

4 

5 

6 

7 

7 

8 

72 

•9511 

.9520 

.9528 

•9537 

•  9546 

•9555 

3 

4 

4 

5 

6 

.  7 

8 

73 

•  9563 

•9572 

•  9580 

•  9588 

•9596 

.9605 

3 

4 

•S 

6 

7 

7 

74 

.9613 

.9621 

.9628 

.9636 

.9644 

.9652 

3 

4 

S 

5 

6 

7 

75 

.9659 

.9667 

.9674 

.9681 

.9689 

.9696 

3 

4 

4 

S 

& 

7 

76 

•9703 

.9710 

.9717 

.9724 

•9730 

•9737 

3 

3 

4 

5 

5 

6 

77 

•  9744 

•9750 

•9757 

•9763 

.9769 

•9775 

3 

3 

4 

4 

5 

6 

78 

.978i 

.9787 

•9793 

•9799 

•  9805 

.9811 

3 

3 

4 

..  S 

5 

79 

.9816 

.9822 

.9827 

•9833 

•  0838 

•9843 

3 

3 

4 

4 

5 

80 

.9848 

•9853 

•  9858 

.9863 

.9868 

.9872 

2 

3 

3 

4 

4 

81 
82 

•9877 

.  99O3 

.9881 

.  99O7 

.9886 

.  99H 

.9890 

.  9914 

•9894 
.9918 

.9899 

.  9922 

2 

3 

3 

3 

4 

83 

•9925 

.9929 

•9932 

•  9936 

•9939 

.9942 

2 

2 

2 

3 

3 

84 

•9945 

.9948 

•9951 

•9954 

•9957 

•9959 

I 

2 

2 

2 

3 

85 
86 

.9962 

.9964 

rv"i*7& 

.9967 
0080 

.9969 
0081 

.9971 

Qng  •} 

•9974 
008  <; 

I 

I 

a 

2 

2 

87 

•997^ 

.9986 

.  997° 

.9988 

.  yyou 
.9989 

•  yy01 
.9990 

•  yyoo 
.9992 

•  yy°j 
•9993 

I 

I 

i 

I 

I 

88 

•9994 

•9995 

.9996 

•9997 

•  9997 

.9008 

0 

O 

o 

I 

I 

89 

.9998 

•9999 

•9999 

.0000 

.0000 

.0000 

o 

o 

0 

0 

0 

O 

o 

O 

0 

bib 
Q 

o' 

10' 

20' 

30' 

40' 

So' 

x' 

a' 

3' 

4' 

5' 

6' 

-7? 

8' 

9' 

Minutes. 

Proportional  parts. 

44 


AZIMUTH 


TABLE  V.  —  LOGARITHMS  OF   NUMBERS. 


8 

Proportional  parts 

.  o 

I 

2 

_ 

4 

5 

6 

M 

8 

g 

I 

2 

3 

4 

5 

6 

7 

8 

9 

I0  .0000 

•0043 

.0086 

.0128 

.0170 

.0212 

•0253 

.0294 

•0334 

•0374 

'  4 

8 

12 

21 

2  5 

20 

33 

37 

II  -0414 

•0453 

.0492 

•0531 

.0569 

.0607 

.0645 

.0682 

.0719 

•0755 

4 

8 

II 

5 

10 

23 

2( 

30 

34 

12  -0792 

.0828 

.0864 

.0899 

•  0934 

.0969 

.1004 

.1038 

.1072 

.  1106 

3 

7 

IO 

4 

17 

21 

24 

28 

3J 

13  -"39 

•1173 

.1206 

.1239 

.1271 

•  1303 

.1335 

•1367 

•1399 

.1430 

3 

ft 

10 

3 

1  6 

10 

23 

2ft 

20 

14  .  1461 

.1492 

•1523 

•1553 

.1584 

.1614 

.1644 

•1673 

•1703 

•1732 

3 

(» 

9 

2 

15 

18 

21 

24 

27 

15  .1761 

.1790 

.1818 

.1847 

•1875 

.1903 

•1931 

•1959 

.1987 

.2014 

3 

6 

8 

ii 

14 

17 

20 

22 

25 

16  -2041 

.2068 

.2095 

.  2122 

.2148 

•  2175 

.2201 

.2227 

•  2253 

.2279 

3 

5 

8 

ii 

13 

1  6 

1  8 

21 

24 

17  -2304 
18  •  2553 

•  2330 

•2577 

.  2601 

.2380 
.2625 

•  2405 
.2648 

.2430 
.2672 

•2455 

.2695 

.2480 
.27l8 

•  2504 
.2742 

•  2529 
•  2765 

5 

5 

7 
7 

10 

9 

12 

15 
14 

17 

2O 
10 

22 
21 

19  .2788 

.2810 

•2833 

.2856 

.2878 

.2900 

.2923 

•2945 

.2967 

.2989 

4 

7 

9 

I  1 

13 

1  6 

1  8 

2O 

20  -3010 

•  3032 

•3054 

•3075 

•  3096 

•  3118 

•3139 

.3160 

.3181 

•  3201 

4 

6 

8 

I  I 

13 

,- 

,- 

10 

21  -3222 

•3243 

•3263 

.3284 

•3304 

•  3324 

•3345 

•3365 

•3385 

•3404 

4 

6 

8 

10 

12 

M 

ift 

1  8 

22  .3424 

•3444 

•3464 

•3483 

•3502 

•3522 

•3541 

•3560 

•3579 

•3598 

4 

ft 

8 

IO 

12 

'4 

i  5 

T7 

23  .3617 

•3636 

.3655 

•3674 

.3692 

•37" 

•3729 

•3747 

.3766 

•3784 

4 

6 

7 

0 

II 

13 

15 

17 

24  .3802 

.3820 

•3838 

.3856 

•3874 

•3892 

•3909 

•3927 

•3945 

•  3962 

4 

5 

7 

9 

II 

12 

'4 

16 

25  -3979 

•3997 

.4014 

.4031 

.4048 

-4065 

.4082 

•4099 

.4116 

•4133 

3 

5 

7 

() 

10 

12 

14 

15 

26.4150 

.4166 

•4183 

.4200 

.4216 

•4232 

.4249 

•4265 

.4281 

.4298 

3 

5 

7 

8 

10 

II 

13 

15 

27  -43J4 

•4330 

•4346 

.4362 

•4378 

•4393 

.4409 

•4425 

.4440 

•4456 

3 

3 

ft 

8 

9 

II 

13 

14 

28  -4472 

.4487 

.4502 

.4518 

•4533 

•4548 

•4564 

•4579 

•4594 

.4609 

3 

5 

ft 

8 

9 

II 

12 

'4 

29  .4624 

•4639 

.4654 

.4669 

•  4683 

.4698 

•4713 

.4728 

•4742 

•4757 

3 

4 

6 

7 

9 

IO 

12 

13 

30  -4771 

.4786 

.4800 

.4814 

.4829 

•4843 

•4857 

.4871 

.4886 

.4900 

3 

4 

6 

7 

9 

IO 

II 

13 

31  -49I4 

.4928 

.4942 

•4955 

•4969 

•4983 

•4997 

.5011 

.5024 

•5038 

3 

4 

6 

7 

8 

10 

II 

12 

32  -5051 

•5065 

•5079 

.5092 

•  5105 

•5"9 

•  5132 

•5145 

•5159 

•5172 

3 

4 

5 

7 

8 

9 

II 

12 

33  -5185 

.5198 

.5211 

•5224 

•5237 

•5250 

•5263 

•5276 

.5289 

•5302 

3 

4 

j 

ft 

8 

9 

IO 

12 

34  -5315 

.5328 

•5340 

•5353 

•5366 

•5378 

•5391 

•5403 

.5416 

•  5428 

3 

4 

5 

ft 

8 

9 

10 

II 

35  -5441 

•5453 

•5465 

•5478 

•5490 

•5502 

•5514 

.5527 

•5539 

•5551 

2 

4 

5 

ft 

7 

0 

10 

„ 

36  -5563 

•5575 

.5587 

•5599 

•  5611 

•5623 

•5635 

•5647 

.5670 

3 

4 

5 

ft 

7 

8 

!O 

TI 

37  .  5682 

•5694 

•5705 

•5717 

•5729 

•5740 

•5752 

•5763 

•5775 

.5786 

3 

5 

ft 

7 

8 

0 

IO 

38  •  5798 

.5809 

.5821 

•5832 

•5843 

.5866 

•5877 

.5888 

•5899 

3 

5 

ft 

7 

8 

9 

10 

39  -59" 

•5922 

•5933 

•5944 

•5955 

.5966 

•5977 

•  5988 

•5999 

.6010 

3 

4 

5 

7 

8 

c 

!O 

40  .6021 

•  6031 

.6042 

•6053 

.6064 

•6075 

.6085 

.6096 

.6107 

.6117 

5 

4 

5 

6 

8 

( 

:o 

41  .6128 

-6138 

.6149 

.6160 

.6170 

.6180 

.6191 

.6201 

.6212 

.6222 

3 

4 

5 

6 

7 

8 

9 

42  .6232 

.6243 

•6253 

.6263 

•  6274 

.6284 

.6294 

•6304 

•  6314 

•6325 

3 

4 

5 

6 

7 

8 

9 

43  -6335 

•6345 

•6355 

•6365 

•6375 

-6385 

•  6395 

.6405 

•6415 

.6425 

3 

4 

5 

6 

7 

8 

9 

44  -6435 

•6444 

•6454 

.6464 

.6474 

.6484 

•  6493 

•6503 

•  6513 

•  6522 

3 

4 

5 

6 

7 

8 

9 

45  -6532 

•6542 

•6551 

-6561 

•6571 

•  6580 

.6<?90 

•6599 

.6609 

.6618 

3 

4 

5 

6 

7 

8 

9 

46  .6628 

•6637 

.6646 

.6656 

.6665 

•6675 

.6684 

•  6693 

.6702 

.6712 

3 

4 

5 

ft 

7 

- 

8 

47  .6721 

•6730 

•  6739 

•6749 

.6758 

.6767 

.6776 

•  6785 

•6794 

.6803 

3 

4 

5 

5 

ft 

7 

8 

48  .6812 

.6821 

.6830 

.6839 

.6848 

.6857 

.6866 

•6875 

.6884 

.6893 

3 

4 

4 

5 

ft 

7 

8 

49  .6902 

.6911 

.6920 

.6928 

•6937 

.6946 

•0955 

.6964 

.6972 

.6981 

3 

4 

4 

5 

ft 

7 

8 

50  .6990 

.6998 

•  7oo7 

.7016 

.7024 

•7033 

.7042 

•  7050 

•7059 

.7067 

3 

3 

4 

5 

ft 

7 

8 

51  .7076 

.7084 

•7093 

.7101 

.7110 

.7118 

.7126 

•7135 

•7143 

•7152 

3 

3 

4 

5 

ft 

7 

8 

52  -7160 

.7168 

.7177 

•7185 

•7193 

.7202 

.  7210 

.7218 

.  7226 

•7235 

2 

3 

4 

5 

ft 

7 

7 

53  -7243 

•7251 

•7259 

.7267 

•  7275 

•7284 

.7292 

•7300 

.7308 

2 

3 

4 

5 

ft 

6 

7 

54  •  7324 

•7332 

•7340 

•7348 

•7356 

•7364 

•7372 

•7380 

•7388 

•7306 

2 

3 

4 

5 

ft 

6 

7 

TABLES 


45 


TABLE   V.  —  (Continued). 


^c 

Proportional  parts. 

£ 

o 

I 

2 

3 

4 

5 

5 

7 

8 

9 

0 

x^ 

I 

2 

3 

4 

5 

6 

7 

8 

9 

.7404 

.7412 

.7419 

•7427 

•7435 

•7443 

•7451 

•  7459 

.7466 

•7474 

, 

2 

2 

6 

7 

# 

.7482 

.7490 

•7497 

•7505 

•7513 

•7520 

•7528 

.7536 

•7543 

•7551 

2 

2 

4 

6 

7 

-7559 

.7566 

•7574 

.7582 

.7589 

•  7597 

.7604 

.7612 

.7619 

.7627 

i 

2 

2 

6 

7 

M 

•7634 

.7642 

.7649 

.7657 

.7664 

.7672 

.7679 

.7686 

.7694 

.7701 

i 

2 

i 

6 

7 

59 

.7709 

.7716 

•7723 

•7731 

•7738 

•7745 

•7752 

,7760 

.7767 

•7774 

i 

2 

3 

4 

6 

7 

60 

.7782 

.7789 

.7796 

.7803 

.7810 

7818 

.7825 

•  7832 

•7839 

.7846 

i 

2 

3 

4 

6 

6 

61 

.7853 

.7860 

.7868 

.7875 

7882 

7889 

.7896 

•  7903 

.7910 

•7917 

i 

2 

3 

/ 

6 

6 

62 

7924 

•  7931 

.7938 

7945 

7952 

7959 

.7966 

•  7973 

.798o 

.7987 

i 

2 

- 

6 

6 

6;-; 

7993 

.8000 

.8007 

8014 

8021 

8028 

8035 

.8041 

.8048 

.8055 

i 

2 

j 

3 

5 

6 

64 

8062 

.8069 

•  8075 

8082 

8089 

8096 

8102 

.8109 

.8116 

.8122 

i 

2 

3 

3 

6 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

.8176 

.8182 

.8189 

i 

2 

^ 

3 

4 

6 

£ 

8i95 

8202 

8209 

8215 

8222 

8228 

8235 

.8241 

8248 

•  8254 

I 

\ 

j 

4 

6 

>7 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

•  8319 

i 

'-. 

3 

4 

6 

,S 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

837° 

8376 

•  8382 

i 

- 

j 

4 

4 

6 

6c> 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

•8445 

i 

2 

3 

4 

4 

6 

7° 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

.8506 

i 

2 

3 

4 

4 

5 

6 

71 

8513 

8519 

852=; 

8531 

8537 

8543 

8549 

8555 

8561 

.8567 

i 

2 

3 

4 

4 

5 

5 

72 

8573 

3?79 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

.8627 

i 

2 

j 

4 

4 

5 

S 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

.8686 

i 

2 

4 

4 

5 

5 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

•  8745 

i 

2 

3 

4 

4 

5 

5 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

.8802 

i 

2 

. 

3 

4 

5 

S 

6 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

•  8859 

i 

2 

3 

4 

5 

5 

7 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

•  8915 

T 

2 

3 

4 

4 

5 

S 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

.8971 

I 

2 

3 

4 

4 

5 

o 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

.9025 

I 

2 

3 

4 

4 

S 

o 

9031 

9036 

9042 

9°47 

9053 

9058 

9063 

9069 

9074 

.9079 

I 

j 

3 

4 

4 

5 

Si 

9085 

9090 

9006 

9101 

9106 

9112 

9117 

9122 

9128 

•9i33 

I 

2 

3 

4 

4 

S 

82 

9138 

9M3 

9149 

9154 

9159 

9165 

9170 

9i75 

9180 

.9186 

I 

2 

3 

4 

4 

5 

S 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

•  9238 

1 

2 

3 

4 

4 

s 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

.9289 

I 

2 

3 

4 

4 

5 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

•934° 

I 

2 

3 

4 

4 

5 

6 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

•939° 

I 

2 

3 

4 

4 

5 

87 

9395 

.9400 

9405 

9410 

9415 

9420 

9425 

943° 

9435 

.9440 

O 

2 

3 

3 

4 

4 

s 

9445 

•  9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

.9489 

0 

2 

3 

3 

4 

4 

80 

9494 

•  9499 

95°4 

95°9 

9513 

95i8 

9523 

9528 

9533 

.9538 

o 

2 

2 

3 

3 

4 

4 

00 

9542 

•9547 

9552 

9557 

9562 

9566 

957i 

9576 

958i 

•  9586 

o 

2 

2 

3 

3 

4 

4 

01 

9590 

•9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

•9633 

0 

2 

2 

3 

3 

4 

4 

02 

9638 

•  9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

.9680 

o 

2 

2 

3 

3 

4 

4 

3 

9685 

.9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

.9727 

0 

2 

2 

3 

3 

4 

4 

04 

9731 

•9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

•9773 

o 

2 

2 

3 

3 

4 

4 

05 

9777 

.9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

.9818 

o 

2 

2 

3 

3 

4 

4 

X> 

9823 

.9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

•  9863 

0 

2 

2 

3 

3 

4 

4 

07 

9868 

.9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

.9908 

0 

2 

2 

3 

3 

4 

4 

>8 

9912 

.9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

•9952 

0 

2 

2 

3 

3 

4 

4 

00 

9956 

.9961 

99<55 

9969 

9974 

9978 

9983 

9987 

9991 

•9996 

0 

2 

2 

3 

3 

3 

4 

46  AZIMUTH 

TABLE   VI. — LOGARITHMIC   VERSED   SINES. 


1 

Minutes. 

Proportional  parts. 

j 

o' 

10' 

2O' 

30' 

40' 

50' 

i 

a' 

3' 

4 

5 

t> 

7' 

»'   9' 

10° 

8.1816 

8.1959 

8.2IOC 

8.  =239 

8.2375 

8.2510 

14 

27 

41 

55 

69 

82 

96 

i  10  123 

II 

.2642 

.2772 

.2900 

.3027 

•  3151 

•3274 

12 

25 

38 

50 

63 

75 

88  100  113 

12 

•3395 

•3514 

.3632 

.3748 

•3863 

•3976 

12 

23  |35 

46 

58 

69 

81 

92  104 

13 

.4087 

.4198 

.4414 

.4520 

.4625 

11 

21  32 

43 

53 

64 

75 

85   96 

14 

.4728 

.4830 

•493 

•5031 

•513° 

.5228 

10 

20 

30 

40 

5° 

60 

70   80   89 

IS 

•5324 

.5420 

•5514 

•5607 

.5700 

•579 

" 

19 

28 

37 

46 

56 

65   74   84 

16 

•  5881 

•  5971 

.6055 

.6147 

•  6234 

.6319 

0 

\- 

2t> 

35 

44 

52 

61   70   79 

17 

.6404 

.6488 

.6572 

.6654 

.6736 

.6817 

8 

16 

25 

33 

49 

57   66   74 

18 

.6897 

.6976 

•7055 

•7133 

.7210 

.7287 

8 

IS 

23  131 

39 

46  |  54   62  |  70 

19 

.7362 

.7438 

•751 

•7586 

•7659 

•7732 

7 

IS 

22 

30 

37 

44 

52 

59 

66 

20 

•7804 

•7875 

•794^ 

.8016 

.8086 

•8155 

7 

14 

21 

28 

35 

42 

49 

56 

63 

21 

.8223 

.8291 

-8358 

•8425 

.8491 

•855- 

7 

i3 

20 

27 

33 

40 

47 

53 

60 

22 

.8622 

.8687 

•875 

•  8815 

.887^ 

.894 

6 

13 

19 

25 

32 

38 

44   5i 

57 

23 

•  9003 

.9065 

.9127 

.9188 

.9248 

.9308 

6 

12 

18 

24 

30 

36   43   49   55 

24 

.9368 

.9427 

.9486 

•9544 

.9602 

.0660 

0 

12 

17 

23 

29 

35 

41 

47   5-2 

25 

8.9717 

8.9774 

8.9830 

8.9886 

8.9942 

8-9997 

6 

IZ 

i7 

22 

28 

34 

39 

45 

50 

26 
27 

9-0052 
•0374 

9.0107  9.016 
.  0426  .  0475 

9.0215 
•0530 

9.0268 
.0582 

19.0321 
.0633 

5 
5 

II 
IO 

16 
16 

21 

21 

27 
26 

32 

38 
36 

43 
41 

48 
47 

28 

.0684 

•  0734 

.0785 

•  0834 

.0884 

•0933 

5 

10 

15 

2O 

25 

30 

35 

40 

45 

29 

.0982 

.1031 

.1079 

.1128 

•"75 

.1223 

5 

IO 

14 

TO 

24 

29 

34 

38 

43 

30 

.1270 

•  1317 

.1364 

.1410 

•1457 

•1503 

5 

0 

'4 

19 

23 

28 

32 

37 

42 

31 

.1548 

•  1594 

.1684 

.1728 

•1773 

4 

9 

13 

18 

22 

27 

3i 

36 

40 

32 

.1817 

.1861 

.1905 

.1948 

.1991 

.2034 

4 

0 

13 

i  - 

22 

26 

30 

35 

39 

33 

.2077 

.2120 

.2162 

.2204 

.2246 

.2288 

4 

8 

13 

17 

21 

25 

29 

34 

38 

34 

•  2329 

•  2370 

.2411 

•2452 

•2493 

•2533 

4 

8 

12 

1  6 

20 

24 

28 

33 

37 

35 

•2573 

.2613 

•2653 

.2692 

.2732 

.2771 

4 

8 

12 

16 

20 

24 

28   32 

36 

36 

.2810 

.2849 

.2887 

.2926]  .2964 

.3002 

4 

8 

II 

15 

19 

23 

27 

31 

34 

37 

.3040 

•  3077 

•3115 

.3152  -3189 

.3226 

4 

7 

II 

15 

19 

22 

26 

30  ]  33 

38 

•  3263 

•3300 

•3336 

•3372  -3409 

•3444 

4 

7 

II 

14 

18 

22 

25 

29 

33 

39 

•3480 

•3516 

•3551 

•3586 

.3622 

.3657 

4 

7 

II 

14 

18 

21 

25 

28 

32 

40 

.3691 

•  3726 

.3760 

•3795 

•3829 

•3863 

3 

7 

10 

14 

17 

21 

24 

27 

3i 

.3897 

•3931  -3964 

•3998 

.4031 

.4064 

3 

7 

10 

'3 

i? 

20 

23 

27 

30 

42 

.4097 

.4130 

.4162 

•4195 

.4227 

.4260 

3 

7 

IO 

16 

20 

23 

26 

29 

43 

.4292 

•  4324 

•4356 

.4387 

.4419 

•4450 

3 

6 

10 

13 

16 

19 

22 

25 

29 

44 

.4482 

•  4513 

•4544 

•4575 

.4606 

.4637 

3 

6 

9 

12 

15 

19 

22 

25 

28 

45 

.4667 

.4698 

•4728 

•4758 

•4788 

.4818 

3 

6 

9 

12 

is 

18 

21 

24 

27 

46 

.4848 

.4878 

.4907 

•4937 

.4966 

•4995 

3 

6 

9 

12 

18 

21 

24 

26 

47 

.5024 

•  5053 

•  5082 

.5140 

-5i68 

3 

6 

9 

I  I 

14 

17 

2O 

23 

26 

48 

•5197 

•  5225 

•5253 

;£& 

•5309 

•5337 

3 

6 

8 

11 

14 

17 

20 

22 

25 

49 

•5365 

•5393 

.5420 

•5448 

•5475 

•5502 

3 

5 

8 

I  I 

14 

16 

19 

22 

25 

50 

•5529 

•5556 

.5583 

•  5610 

•5637 

•5663 

3 

5 

8 

II 

13 

16 

19 

21 

24 

Si 
52 

•5690 
•5847 

-5716 
.5873 

•5743 
•5899 

.5769 
•5924 

•5795 
•5950 

•  5821 
•5975 

3 
3 

5 
5 

8 
8 

IO 
10 

13 
13 

16 
15 

18 
18 

21 

21 

24 
23 

53 

.6001 

.6026 

•  6051 

.6076 

.6101 

.6126 

3 

5 

8 

IO 

13 

IS 

18 

20 

23 

54 

.6151 

.6176 

.6201 

.6225 

•  6250 

.6274 

2 

5 

7 

o 

12  ' 

IS 

J7 

20 

22 

55 
56 

.6298 
.6442 

•  6323 
.6466 

•6347 
.6490 

•6371 
•6513 

.6395 
•6537 

.6419 
.6560 

5 
5 

7 
7 

IO 

D 

12 
12 

14 

li 

19 
19 

22 
21 

57 

•  6584 

.6607 

.6630 

•6653 

.66-76 

.6699 

5 

7 

o 

II 

14 

16 

18 

21 

58 

.6722 

.6744 

.6767 

.6790 

.6812 

•6835 

5 

7 

0 

II 

14 

16 

18 

20 

59 

•6857 

.6879 

.6902 

.6924 

.6946 

.6968 

4 

7 

0 

II 

13 

IS 

18 

20 

TABLES 

TABLE    VI.  —  (Continued). 


47 


bb 
Q 

Minutes. 

Proportional  parts. 

o' 

10' 

20' 

30' 

40' 

So' 

l' 

2f 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

60° 

9.6990 

9.7012 

9-7033 

9-7055 

9.7077 

9.7098 

4 

6 

9 

ii 

13 

15 

17 

20 

61 

.7120 

.7141 

.7162 

.7184 

•7205 

.7226 

4 

0 

8 

ii 

13 

15 

17 

19 

62 

.7247 

.7268 

•7289 

•  7310 

•  7331 

•  7351 

4 

6 

8 

10 

12 

15 

17 

19 

63 

•7372 

•7393 

•7413 

•7434 

•7454 

•7474 

4 

6 

8 

10 

12 

14 

16 

18 

64 

•7494 

•7515 

•7535 

•7555 

•7575 

•7595 

4 

6 

8 

IO 

12 

14 

16 

18 

65 

•7615 

•7634 

.7654 

.7674 

•7693 

•7713 

4 

6 

8 

IO 

12 

14 

16 

18 

66 

•7732 

•7752 

.7771 

.7791 

.7810 

.7829 

4 

6 

8 

10 

12 

'14 

15 

i7 

67 

.7848 

.7867 

.7886 

•7905 

.7924 

•7943 

4 

6 

8 

9 

II 

13 

15 

i? 

68 

.7962 

.7980 

•7999 

.8017 

8036 

-8054 

4 

6 

7 

9 

II 

13 

15 

i? 

69 

.8073 

.8091 

.8110 

.8128 

.8146 

.8164 

4 

s 

7 

9 

II 

13 

IS 

16 

70 

.8182 

.8200 

.8218 

.8236 

•8254 

.8272 

4 

5 

7 

9 

II 

13 

14 

16 

?i 

.8289 

.8307 

•8325 

-8342 

.8360 

•8377 

4 

5 

7 

9 

II 

12 

T4 

16 

72 

•8395 

.8412 

.8429 

•8447 

.8464 

.8481 

3 

5 

7 

9 

IO 

12 

14 

15 

73 

.8498 

•8515 

•8532 

•8549 

.8566 

-8583 

3 

5 

7 

8 

10 

12 

14 

15 

74 

.8600 

.8616 

•8633 

.8650 

.8666 

.8683 

3 

5 

7 

8 

ID 

12 

13 

15 

75 

.8699 

.8716 

.8732 

.8748 

•8765 

.8781 

3 

5 

7 

8 

10 

II 

13 

15 

76 

.8797 

•  8813 

.8829 

.8845 

.8861 

•8877 

3 

S 

6 

8 

10 

II 

13 

14 

77 

.8909 

•  8925 

.8941 

•  8956 

.8972 

3 

5 

(> 

8 

9 

II 

13 

14 

78 

.8988 

.9003 

.9019 

•9034 

.9050 

•  9065 

3 

5 

6 

8 

9 

II 

12 

14 

79 

.9081 

r9096 

.9111 

.9126 

.9141 

•9157 

3 

5 

6 

8 

9 

II 

12 

14 

80 

.9172 

.9187 

.9202 

.9217 

.9232 

.9246 

3 

4 

6 

7 

9 

IO 

12 

13 

81 

.9261 

.9276 

.9291 

•9305 

•  9320 

•9335 

3 

4 

0 

7 

9 

10 

12 

13 

82 

•9349 

•9364 

•9378 

•9393 

.9407 

•9421 

3 

4 

6 

7 

9 

10 

12 

T3 

83 

•9436 

•945° 

.9464 

•9478 

.9492 

•95o6 

3 

4 

6 

7 

9 

10. 

II 

13 

84 

•9521 

•9535 

•9548 

•  9562 

•  9576 

•9590 

3 

4 

6 

7 

8 

10 

II 

13 

85 

.9604 

.9618 

.9631 

•9645 

•9659 

.9672 

3 

4 

5 

7 

8 

10 

II 

12 

86 

.9686 

•9699 

•9713 

.9726 

.9740 

•9753 

3 

4 

5 

7 

8 

9 

II 

12 

8? 

•9767 

.9780 

•9793 

.9806 

.9819 

•9833 

3 

4 

5 

7 

8 

9 

II 

12 

88 

.9846 

.9859 

.9872 

•  9885 

.9898 

.9911 

3 

4 

5 

6 

8 

9 

10 

12 

89 

9.9924 

9.9936 

9-9949 

9.9962 

9-9975 

9.9987 

3 

4 

5 

6 

8 

9 

IO 

II 

90 

0.0000 

0.0013 

0.0025 

0.0038 

0.0050 

0.0063 

3 

4 

5 

6 

8 

9 

IO 

II 

9i 

.0075 

.0088 

.0100 

.0112 

•  0125 

•0137 

2 

4 

5 

6 

7 

9 

10 

II 

92 

.0149 

.0161 

•0173 

-0185 

.0197 

.0210 

2 

4 

5 

6 

7 

9 

10 

II 

93 

.0222 

.0234 

.0245 

•0257 

.0269 

.0281 

2 

4 

5 

6 

7 

8 

9 

II 

94 

.0293 

•0305 

.0316 

.0328 

.0340 

•0351 

2 

3 

S 

6 

7 

8 

9 

10 

95 

•  0363 

•0374 

.0386 

•0397 

.0409 

.O42O 

2 

3 

5 

6 

7 

8 

9 

10 

96 

.0432 

•0443 

•  0454 

.0466 

•0477 

.0488 

2 

3 

4 

6 

7 

8 

9 

10 

97 

.0499 

.0511 

.0522 

•°533 

•0544 

•0555 

2 

3 

4 

6 

7 

8 

9 

IO 

98 

.0566 

•0577 

.0588 

•0599 

.0610 

.O62O 

2 

3 

4 

5 

6 

8 

9 

10 

99 

.0631 

.0642 

•0653 

.0663 

.0674 

.0685 

2 

3 

4 

5 

6 

8 

9 

10 

100 

•  0695 

.0706 

.0717 

.0727 

.0738 

.0748 

2 

3 

4 

5 

6 

7 

8 

10 

101 

.0758 

.0769 

.0779 

.0790 

.0800 

.O8l0 

2 

3 

4 

5 

6 

7 

8 

9 

IO2 

.0820 

.0831 

.0841 

.0851 

.0861 

.0871 

2 

3 

4 

5 

6 

7 

8 

9 

103 

.0881 

.0891 

.0901 

.0911 

.0921 

.0931 

2 

3 

4 

5 

6 

7 

8 

9 

104 

.0941 

.0951 

.0961 

.0970 

.0980 

.0990 

2 

3 

4 

5 

6 

7 

8 

9 

IOS 

.1000 

.1009 

.1019 

.1029 

.1038 

.1048 

2 

3 

4 

5 

6 

7 

8 

9 

106 

•  1057 

.  1067 

.  1076 

.1086 

.1095 

•  HO5 

2 

3 

4 

5 

6 

7 

8 

9 

107 

.1114 

.1123 

•  1133 

.1142 

.1151 

.  1160 

2 

3 

4 

5 

6 

6 

7 

8 

108 

.1169 

.1179 

.1188 

.1197 

.1206 

.1215 

2 

3 

4 

5 

5 

6 

7 

8 

109 

.1224 

•1233 

.  1242 

.1251 

.1260 

.1269 

2 

3 

4 

5 

5 

6 

7 

8 

AZIMUTH 


TABLE  VII.  —HOUR  ANGLES  OF  d  CASSIOPEIA. 
Intervals: —  1910,  6ms8s;    1920,  ioms7s;    1930,   i5mi3s. 


Latitudes. 

Alt. 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

32° 

34° 

6° 

io6°.o 

109°.  9 

113°.  8 

n8°.o 

122°.  5 

127°.  3 

132°-  7 

138°.  7     145°.  9 

I55°-  2 

8 

101.8 

105-5 

109.3 

ii3-3 

"7-5 

122.  O 

126.  9 

132.2      138.3 

145-5 

10 

97-7 

IOI.2 

104.9 

108.8 

112.  8 

I.I7.0 

121.5 

126.4      131-8 

137-9 

12 

93-6 

97-1 

100.7 

104.4 

108.2 

112.  2 

116.4 

121.  0        125.9 

14 

89.6 

93-o 

96-5 

100.  I 

103.8 

IO7.6 

in.  6 

115.9  i   120.4 

125-3 

16 

85.6 

89-0 

92-4 

95-9 

99-5 

103.2 

107.0 

III.O 

ii5-3 

II9-8 

18 

81.6 

85-0 

88.4 

91.8 

95-3 

98.9 

IO2.6 

106.4 

110.4 

II4.7 

20 

77-6 

81.0 

84.4 

87-7 

91. 

94-7 

98.2 

101.9 

105.8 

109-8 

22 

73-6 

77-o 

80.4 

83-7 

87- 

90-5 

94-o 

97-6 

101.3 

105  •  r 

24 

69.6 

73-o 

76.4 

79-7 

83- 

86.5 

89-9 

93-3 

06.9 

IOJ.O 

26 

65-6 

69.0 

72.4 

75-8 

79- 

82.4 

85-8 

89-2 

92.6 

96.2 

28 

61.4 

65.0 

68.4 

71.8 

75- 

78.4 

81.7 

85-1 

88.4 

91.9 

30 

60.9 

64.4 

67-8 

74-5 

77-7 

81.0 

84-3 

87.7 

32 

52-9 

56.7 

60.3 

63.8 

67  '. 

70-5 

73-8 

77.0 

80.3 

83.6 

34 

48-4 

52-4 

56-1 

59-7 

63- 

66.5 

69.8 

73-0 

76-3 

79-5 

36 

43-7 

47-9 

51-8 

55-5 

59- 

62.5 

65-8 

69.1 

72-3 

75-5 

38 

38.7 

43-2 

47-4 

5i-3 

54-9 

58.5 

61.8 

65.1 

08.3 

71-5 

40 

33-2 

38-3 

42.8 

46.9 

50-7 

54-3 

57-8 

61.1 

64-3 

67-5 

42 

27.0 

32-9 

37-8 

42.2 

46-3 

50-1 

53-6 

57-o 

60.3 

63-5 

44 

26.7 

32-4 

37-3 

41.7 

45-7 

49-4 

53-o 

56-3 

59-6 

46 

26.3 

32-0 

30-9 

41.2 

48.8 

52-2 

55-5 

48 

26.  o 

31  6 

36.3 

40  6 

44-  5 

48.1 

51  *  5 

50 

25-6 

31.2 

35-8 

40.0 

43-8 

47-3 

25.  2 

so.  6 

35-  2 

39-  3 

AT..  T 

54 



ow*  ^ 

24-8 

30.1 

34-7 

*tO'  * 

38-7 

Latitudes. 

Alt. 

36° 

38° 

40° 

42° 

44° 

46° 

48° 

50° 

52° 

54° 

8° 

10 

154°  -9 
145.2 

154.6 
144.8 

154.  4 

j-70     H 

154-  o 

16 

130.0 
124.8 

130-3 

136^5 

144.0 

153-7 

18 

119.2 

124.2 

129.7 

136.0 

143.6 

153-4 

20 

114.0 

118.6 

123.6 

129.2 

135-5 

143-1 

153-0 

22 

109.1 

II3-4 

n8.o 

123.0 

128.6 

135-0 

142.6 

152-7 

24 

104.4 

108.4 

112.  7 

117-3 

122.3 

127.9 

134-4 

142.1 

152-2 

26 

99-8 

103-7 

107.7 

112.  0 

116.6 

121.  6 

127.2 

133-7 

I4L5 

151-8 

28 

95-4 

99-1 

102.9 

106.9 

III.  2 

115-8 

120.9 

126.5 

133-0 

140.9 

30 

M.I 

94.6 

98.3 

102.  I 

106.1 

110.4 

115-0 

I2O.O 

125-7 

132-3 

32 

86.9 

90-3 

93-8 

97-4 

IOI.2 

105.2 

109.5 

II4.I 

119.2 

124.9 

34 

82.8 

86.1 

89-4 

92-9 

96.5 

100.3 

104.3 

I08.5 

113-1 

118.2 

36 

78-7 

81.9 

85-2 

88.5 

92.0 

95-5 

99-3 

103-3 

107-5 

112.  I 

38 

74-7 

77-8 

81.0 

84-3 

87.6 

91.0 

94-5 

98.2 

102.2 

I06.4 

40 

70.7 

73-8 

76.9 

80.  i 

83.3 

86.5 

89-9 

93-4 

97.1 

IOI.O 

42 

66.7 

69.8 

72-9 

75-9 

79-o 

82.2 

85-4 

88.7 

92.2 

95-8 

62.7 

65-8 

68.8 

71.9 

74-9 

77-9 

81.0 

84-3 

87-5 

90.9 

46 

58.7 

61.8 

64-8 

67-8 

70.8 

73-8 

76.8 

79-8 

82.9 

86.1 

48 

54-7 

57-8 

60.9 

63-8 

66.8 

69-7 

72.6 

75-5 

78-4 

81.4 

50 

50.7 

53-8 

56-9 

59-9 

62.8 

65.6 

68.4 

71.2 

74.1 

76.9 

52 

46-5 

49-8 

52-9 

55-9 

58.8 

61.6 

64-3 

67-1 

69.8 

72-5 

54 

42-3 

45-7 

48.9 

51-9 

54-8 

57-6 

60.3 

63.0 

65-6 

68.2 

56 

37-9 

4i-5 

44-8 

47-9 

50.8 

53-6 

56-3 

58-9 

6l.S 

63-9 

58 

33-3 

37-2 

40.6 

43-8 

46.8 

49.6 

52-3 

54-9 

57-4 

59-8 

60 

28.4 

32-6 

36-3 

39-7 

42.8 

45-6 

48-3 

50-9 

53-3 

55-7 

NOTE.  —  If  the  star  is  east  of  the  meridian  subtract  this  hour  angle  from  360°. 


TABLES 

TABLE    VIII.— COORDINATES    OF    POLARIS. 


49 


4 

1910 

1920 

1930 

^ 

^ 

I9IO 

1920 

1930 

< 

E 

P 

£ 

P 

t 

p 

P 

E 

w 

P 

* 

P 

P 

P 

P 

E 

sin  t 

COS    / 

sin  t 

cos  t 

sin  t 

cos  t 

sin  t 

COS/ 

sin  / 

cost 

sin  / 

cost 

30 

32 

35-2 
37-3 

61.0 

33-7 
35-7 

58.3 

57-1 

32.1 
34-1 

55-6 

54-5 

150 

148 

60 
62 

61.0 
62.2 

35-2 

33-  1 

58-3 
59-5 

33-7 
31-6 

55-6 
1  56.7 

32.1 
30.2 

120 

118 

34 

39-4 

58.4 

37-7 

55-8 

35-9 

53-3 

146 

64 

63-3 

3°-9 

60.5 

29-5 

57-8 

28.2 

116 

36 

41.4 

57-o 

39-6 

54-5 

37-8 

52.0 

144 

66 

64.4 

28.6 

61-5 

27.4 

58.7 

26.1 

114 

38 

43-4 

55-5 

53-  i 

39-6 

50.6 

142 

68 

65-3 

26.4 

62.4 

25-2 

59-6 

24.1 

112 

40 

45-3 

54-o 

43-3 

51-6 

41-3 

49-2 

140 

70 

66.2 

24.1 

63-3 

23-0 

60.4 

22.0 

no 

42 

47-i 

52.4 

45-1 

50.0 

43-o 

47-7 

138 

72 

67.0 

21.8 

64.1 

20.8 

61.1 

19.9 

108 

44 

48.9 

50-7 

46.8 

48.4 

44-6 

46.2 

136 

74 

67.7 

19.4 

64.7 

18.6 

61.8 

17.7 

106 

46 

50.7 

48.9 

48.4 

46.8 

46.2 

44.6 

134 

76 

68.3 

17.0 

65-3 

16.3 

62.3 

15-5 

104 

48 

52-4 

47.1 

50.0 

45-i 

47-7 

43-o 

132 

68.9 

14.6 

14.0 

62.9 

13-4 

102 

50 

54-o 

45-3 

51-6 

43-3 

49-2 

41-3 

130 

80 

69.4 

12.2 

66.1 

11.7 

63-3 

II.  2 

100 

52 

55-5 

43-4 

53-i 

41-5 

50.6 

39-6 

128 

8a 

69-8 

9.8 

66.7 

9-4 

63.6 

8.9 

98 

54 

57-o 

41.4 

54-5 

39-  ^ 

52.0 

37-8 

126 

84 

70.1 

7-4 

67-0 

7.0 

63-9 

6.7 

96 

56 

58-4 

39-4 

55-8 

37-7 

53-3 

35-9 

124 

86 

70.3 

4-9 

67.2 

4-7 

64.1 

4-5 

94 

58 

59-7 

37-3 

57-i 

35-7 

54-5 

34-1 

122 

88 

70.4 

2-5 

67-3 

2.4 

64.2 

2.2 

92 

60 

61.0 

35-2 

58.3 

33-7 

55-6    32-  i 

120 

90    70.4 

0.0 

67.4    o.o 

64-3 

O.O 

90 

TABLE    IX.  —  CORRECTION    FOR    AZIMUTH. 

p  sin  t. 

Proportional  parts. 

Alt. 

30' 

40' 

So' 

60 

70' 

i' 

~ 

2' 

3' 

4' 

5' 

6' 

7' 

8' 

9' 

15° 

I* 

.  i 

i'-4 

; 

'.8 

2'  .  i 

2'.  5 

/ 

.'i 

.'j 

.'2 

.'2 

.'2 

'3 

.'3 

18 

i 

-S 

2.1 

2.6 

3.1 

3-6 

.2 

.  2 

•3 

•3 

•4 

•4 

•5 

21 

1 

.X 

2.8 

3-6 

4-3 

5-o 

.2 

•  3 

•4 

•4 

•5 

.6 

.6 

24 

•    9 

.8 

3-8 

4-7 

5-7 

6.6 

•3 

•4 

•5 

.6 

•  7 

.8 

•9 

27 

3 

•7 

4-9 

6.1 

7-3 

8.6 

•4 

•5 

.6 

•  7 

-9 

X 

.0 

i.i 

30 

4 

.6 

6.2 

7-7 

9-3 

10.8 

•  3 

•  5 

.6 

.8 

•9 

.1 

.2 

•4 

31 

5 

.0 

6-7 

8-3 

10.  0 

11.7 

•3 

•5 

•7 

.8 

.0 

.  2 

•3 

•  5 

32 

's 

•4 

7-2 

9.0 

10.8 

12.5 

•4 

-5 

•  7 

•9 

.1 

•  3 

•4 

.6 

33 

i 

.8 

7-7 

9.6 

"•5 

13-5 

•4 

.6 

.8 

.0 

.2 

•3 

•5 

•7 

34 

6 

•  4 

8.2 

j 

0.  1 

12.4 

14.4 

•4 

4 

( 

o 

.  2 

6 

35 

6.6 

8.8 

II.  0 

13-2 

15-5 

•4 

•  7 

•9 

.1 

•3 

•5 

.S 

.0 

36 

7 

.  I 

9-4 

i 

1.8 

14.2 

16.5 

-5 

•  7 

•9 

.  2 

•4 

•  7 

•0 

.1 

37 

7 

.6 

10.  I 

i 

2.6 

I5-I 

17-6 

•3 

•5 

.8 

.0 

•3 

-5 

.8 

.0 

•  3 

38 

8.1 
8.6 

10.8 

TT      C 

13-5 
14-  3 

16.1 

17.  2 

18.8 

2O.  I 

•  3 

7 

•  5 
.6 

.8 

r] 

.1 

•3 

.6 

•9 

.  i 

•4 
.6 

39 

40 

9.2 

11  '  3 

12.2 

15-3 

18.3 

21.4 

"  O 

•3 

.6 

•  V 

•Q 

.2 

•  5 

.8 

.  i 

•  3 
4 

•  7 

9.8 

III 

i6.3 

i**  ^ 

19-5 

22.8 

•  3 

•  7 

.c 

•3 

.6 

.0 

•3 

.6 
.8 

•9 

42 
43 

10.  4 

II.  0 

14.7 

i 

/  •  o 

8.4 

22.0 

25-7 

•4 

-7 

.  i 

•  5 

.8 

.2 

•  4 
.6 

-0 

3-  J 
3-3 

44 

ii 

•  7 

15.6 

19-5 

23-4 

27-3 

•4 

.8 

.2 

.6 

.0 

•3 

•  7 

3 

.  I 

3-5 

45 

12 

•4 

16.6 

20.7 

24.9 

29.0 

•4 

.8 

.2 

•  7 

.  i 

•5 

2.9 

3 

•3 

3-7 

46 

13 

.2 

17.6 

2 

2.0 

26.4 

30.8 

•4 

•9 

•  3 

.8 

.2 

.6 

3-  I 

3 

•5 

3-9 

47 

14 

.0 

18.7 

2 

3-3 

28.0 

32.6 

•5 

•9 

-4 

•9 

•3 

.8 

3-3 

3 

•  7 

4.2 

48 

14.8 

19.8 

24-7 

29.7 

34-6 

•5 

.0 

•5 

.0 

•5 

3.0 

3-5 

4 

0 

4-4 

49 

15.7 

21.  O 

26.2 

31-5 

36-7 

•5 

.0 

.6 

.6 

3-1 

3-7 

4 

.  i 

4-7 

5° 

16.7 

22.2 

27-8 

33-3 

38-9 

.6 

.1 

•  7 

.  2 

.8 

3-3 

3-9 

4 

4 

5-o 

T7 

.7 

23-6 

29-5 

35-3 

41.2 

.6 

.  2 

.8 

•4 

•9 

3-5 

4.1 

4 

•7 

5-3 

52 

18 

-7 

25.0 

', 

1.2 

37-5 

43-7 

.6 

.2 

•Q 

•5 

3-J 

3-7 

4-4 

5 

0 

5-6 

53 

19.8 

26.5 

33-1 

39-7 

46-3 

•7 

•3 

.c 

.6 

3-3 

4.0 

4.6 

5 

•3 

6.0 

54 

21 

.0 

28.1 

2 

5.  1 

42.1 

49.1 

-7 

•4 

.  i 

.8 

3-5 

4-2 

4-9 

5 

.6 

6-3 

55 

22 

•  3 

29.7      37.2      44.6      52.0 

•7 

•  5 

.2 

3-0 

3-7 

4-5 

5-2    5 

•Q 

6.7 

5° 


AZIMUTH 


TABLE  X.  —  HOUR  ANGLES  OF  d  DRACONIS. 
Intervals:    1910,  6h  i4m  2is;  1920,  6h  i8m  58s;    1930,  6h  2 


Latitudes. 

Alt. 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

30° 

32° 

34° 

6° 
8 

10 
12 
14 

16 
18 

20 
22 
24 
26 
28 
30 
32 

34 
36 
38 
40 
42 
44 
46 
48 
50 

H4°.i 
108.3 
102.7 
97-3 
92.0 
86.7 
81.5 
76.3- 
71.0 
65-6 
60.0 
54-2 
48.! 
4i-5 
34-1 
25-0 

1  19°  .8 
"3-7 
107.9 
102.3 
96.9 
91.6 
86.3 
81.1 
75-8 
70-5 
65.2 
59-6 
53-9 
47-8 
41.2 
33-8 
24-8 

1  26°.  i 
II9-5 
"3-3 
107-5 
101.9 
96.5 
91.1 
85-9 
80.6 
75-4 
70.1 
64-7 
59-2 
53-5 
47-4 
40.9 
33-5 
24.6 

133°  -0 

125.7 
119.1 
112.9 
107.1 
101.5 
96.0 

90.7 
85-4 
80.2 

74-9 
69.6 
64-3 
58-8 
53-1 
47.0 
40-5 
33-2 
24.4 

140°  .6 
132.6 
125-4 
118.7 
112.5 
106.7 

IOI.O 

95-6 

00.2 

84.9 
79-7 
74-5 
69-2 
63-8 
58-3 
52.6 
46.6 
40.2 
32-9 
24.1 

i5i°-o 
140.7 
132.3 
125.0 
118.3 

112.  1 

106.2 
100.6 
95-i 
89.7 
84.4 
79-2 
74.0 
68.7 
63-4 
57-9 
52.2 
46.2 
39-8 
32-6 
23-9 

0 

"iSO.'s" 
140.4 
132.0 
124.6 
117.9 
in.  6 
105.8 

100.  I 

94-6 
89.2 
83-9 
78.7 
73-5 
68.2 
62.9 
57-4 
51-8 
45-8 
39-4 
32-3 
23-6 

0 



0 

150.5 
140.1 
131-6 
124.2 
"7-5 

III.  2 
105.3 

99-6 
94.1 
88.7 
83-4 
78.2 
72.9 
67.7 
62.4 
56-9 
5i-3 
45-4 
39-o 
3i-9 

139.8 
I3I-3 
123-8 
117.1 
no.  8 
104.8 
09.1 
93-6 
88.2 
82.9 
77-6 

72-4 
66.9 
61.8 
56.4 
50.8 
44-9 
38.6 

139-5 
130.9 
123.4 
116.6 
110.3 
104.3 
98.6 
93-o 
87.6 
82.3 
77.0 
71-8 
66.6 
61.3 
55-9 
50-3 
44.4 

38.1 

Latitudes. 

Alt. 

36° 

38° 

40° 

42° 

44° 

46° 

48° 

50° 

52° 

54° 

18° 

20 
22 
24 
26 
28 
30 
32 

34 
36 
38 
40 
42 
44 
46 
48 
50 
52 

I 

60 

139°  -2 

I30-5 
123.0 
116.2 
109.8 
103.8 
98.0 
92.4 
87.0 

81.7 
76.5 

71.2 
66.0 
60.7 
55-3 
49-7 
43-9 
37-7 

i38-'8" 
130.1 
122.5 
"5-7 
109.3 
103.2 
97-4 
91.8 
86.4 
81.1 
75-8 
70.6 
65-3 
60.0 
54-7 
49.1 
43-3 
37-2 

0 

0 

° 

138.5 
129.7 

122.  I 
H5-I 
108.7 
IO2.6 

96.8 
91.2 

85.7 
80.4 

75-i 
69-9 
64.7 
59-4 
54-o 
48-5 
42.8 
36-6 

138.! 
129.3 

121.  6 

114.6 
108.1 

102.0 
96.2 
9°-5 
85-I 
79.6 

74-4 
69-2 
63-9 
58.7 
53-3 
47-8 
42.1 
36.0 

137-7 
128.8 

121.  0 
II4.0 
107-5 
IOI-3 

95-5 
89.8 
84-3 
78.9 
73-6 
68.4 
63-2 
57-9 
52-6 
47-1 
41.4 
35-4 

137-3 
128.3 
120.5 
"3-4 
106.8 
100.6 
94-7 
89-0 
83-5 
78.1 
72.8 
67.6 
62.3 
57-i 
51.8 
46.3 
40-7 

136.8 
127.8 
119.9 
112.7 
106.1 
99-9 
93-9 
88.2 
82.7 
77-2 
71.9 
66.7 
61.4 
56.2 
50.9 
45-5 

136.4 
127.2 
119.2 

112.  0 
105.3 
99.1 

93-1 
87-3 
81.7 
76-3 
71.0 
65-7 
60.5 
55-2 
49-9 

135-8 
126.6 
118.5 

III.  2 
104-5 
98.2 
92.1 
86.3 
80.7 

75-3 
69-9 
64.6 

59-4 
54-i 

135-3 
125.9 
117.8 
110.4 
103.6 
97-2 
91.1 
85-3 
79-6 
74-i 
68.7 
63-4 
58-2 

NOTE. — If  the  star  is  east  of  the  meridian  subtract  this  hour  angle  from  360°, 


TABLES 


TABLE    XI.  —  COORDINATES    OF    POLARIS. 


I9io 

1920 

1930 

< 

1910 

< 

1920 

1930 

^ 

ffi      p 

P 

P 

P 

P 

P 

K    I 

ij       P 

# 

P 

P 

P 

P 

ffi 

HH 

sin 

t  cos  / 

sint 

cos  / 

sin  /  c 

ost 

sin  t 

COS* 

sin  *  < 

sin  t 

cost 

o    o.< 

>   70.4 

0.0 

67.4 

o.o 

54-3 

180 

5    18.2 

68.0 

17.4 

65.1 

16.6 

62.1 

165 

I    I. 

J    70.4 

1.2 

67.4 

i.i 

54.2 

79 

6    19.4 

67.7 

18.6 

64.7 

17.7 

61.8 

164 

2      2.. 

>   70.4 

2.4 

67.4 

2.  2 

54.2 

[78 

7    20.6 

67.4 

19.7 

64.4 

18.8 

61.5 

163 

3    3- 
4    4-( 

J   70.3 
>  70.2 

3-5 
4-7 

«7.3 

67.2 

3-4 
4-5 

54.2 

54.1 

77 
76 

8    21.8 
9    22.9 

67.0 
66.6 

20.8 

21.9 

64.1 
63-7 

19.9 
20.9 

61.1 
60.8 

162 
161 

S    6. 

70.1 

5-9 

67.1 

5-6 

54.0 

75    '- 

o    24.1 

66.2 

23-0 

63-3 

22.0 

60.4 

160 

6    ?.< 

70.0 

7.0 

67.0 

6.7 

53.9 

74    : 

>i    25.2 

65.8; 

24.1 

62.9 

23.0 

60.0 

159 

7    8.< 

)  69.9 

8.2 

66.9 

7-8 

53.8 

73    - 

2      26.4 

65.3 

25.2 

62.4 

24.1 

59-6 

158 

8    9.* 

5    69.7 

9-4 

66.7 

8.9 

53.6 

72    : 

3    27.5 

64.8 

26.3 

62.0 

25-1 

59-1 

i57 

9  ii.  c 

>    69.5 

10.5 

66.5 

10.  I 

53.5 

[71    '. 

4    28.6 

64.4 

27.4 

61.5 

26.1 

58.7 

156 

IO  12.  i 

69-3 

11.7 

66.3 

II.  2 

53.3 

170    : 

'5     29.8 

63.8 

28.5 

61.0 

27.2 

58.2 

155 

II  13.-: 

t    69.1 

12.8 

66.2 

12-3 

53.1 

169    . 

>6    30.9 

63-3 

29-5 

60.5 

28.2 

57-8 

154 

12   I4-C 

>    68.9 

14.0 

65-9 

13-4 

52.9 

[68 

27    32.0 

62.8 

30.6 

60.0 

29.2 

57-3 

153 

13  is-* 

5    68.6 

65-7 

14.4 

52.6 

[67 

28    33-1 

62.2 

31.6 

59-5 

30-2 

56.7 

152 

14  17.  c 

>    68.3 

i6'.3 

65-4 

15-5 

62.3 

r66 

29    34.2 

61.6 

32-7 

58-9 

3L2 

56.2 

151 

15  18.2    68.0 

17.4    65.1 

16.6    62.! 

165 

?o    35-2 

61.0 

33-7    58.3 

32.1 

55-6 

150 

TABLE    XII.  —  CORRECTIONS    FOR    AZIMUTH. 

p  sin  t. 

Proportional  parts. 

Alt. 

10' 

20' 

30' 

40' 

i' 

2' 

3' 

4' 

5' 

6' 

7 

8' 

9' 

18 

•5 

I.O 

2.1 

.  i 

.  i 

.2 

2 

•3 

•  3 

•4 

'•3 
•  4 

'•3 
•5 

21 

•  7 

1.4 

2.1 

2.8 

.1 

.1 

.2 

3 

•4 

•4 

•5 

. 

6 

.6 

24 

•9 

1.9 

2.8 

3-8 

.1 

.2 

•3 

•4 

•  5 

.6 

•7 

. 

8 

•9 

27 

1.2 

2.4 

3-7 

4.9 

.1 

.2 

•4 

•5 

.6 

•  7 

•9 

I.O 

i.i 

3° 

.5 

3-1 

4.6 

6.2 

.2 

•3 

•  5 

.6 

.8 

•9 

i.i 

2 

1.4 

.7 

g 

3-3 

i  6 

5-o 

6-7 

.  2 

•3 

•5 

•  7 

IV 

.8 

I.O 

1.  1 

1.2 
I    1 

•3 

f  A 

1:1 

32 
33 

•9 

3-° 

3-8 

5-8 

7-7 

.2 

•4 

•  5 
.6 

•   / 

.8 

I.O 

I.  2 

*  •  o 

1-3 

*+ 

5 

34 

.  i 

4.1 

6^2 

8.2 

.2 

•4 

.6 

.8 

I.O 

1.2 

1.4 

.6 

1.9 

g 

.2 

4-4 

6.6 

8.8 

.2 

•4 

•  7 

•9 
.  9 

.  i 

1-3 

1-5 

8 

2.0 

3° 

37 

•  5 

5-o 

7.6 

10.  1 

•3 

•  5 

•5 

!s 

.O 

•3 

i-5 

1.8 

.0 

2-3 

38 

•  7 

5-4 

8,1 

10.8 

•  3 

•  5 

.8 

.1 

•3 

1.6 

1.9 

i 

2-4 

39 

•9 

5-7 

8.6 

n-5 

•3 

.6 

•9 

.1 

•4 

i-7 

2.0 

3 

2.6 

40 

3-1 

6.1 

9.2 

12.2 

•  3 

.6 

•9 

.  2 

•  5 

1.8 

2.  I 

4 

2.7 

3-3 

6-5 
6,, 

9-8 

13-0 
T,    g 

•3 

•  7 

.0 

•3 

.6 

2.0 

2-3 

.6 

.8 

2.9 

42 
43 

3-5 

3-7 

•9 

7-3 

II.  O 

13.0 
14.7 

•3 
•4 

•7 

.1 

!3 

.8 

2.2 

2.6 

•9 

3'  ^ 
3-3 

44 

3-9 

7-8 

11.7 

15-6 

•4 

.8 

.2 

.6 

.0 

2-3 

2.7 

3- 

1 

3-5 

45 

4.1 

8-3 

12.4 

16.6 

•4 

.8 

.2 

.7 

.  i 

2-5 

2.9 

3- 

3 

3-7 

46 

4-4 

8.8 

13-2 

17-6 

•4 

•9 

•  3 

.8 

.2 

2.6 

3-1 

3- 

5 

3-9 

47 

4-7 

9-3 

14.0 

18.7 

•5 

•9 

•4 

•Q 

.3 

2.8 

3-3 

3- 

7 

4.2 

48 

4-9 

9-9 

14.8 

19.8 

•  5 

s  c 

.0 

•5 

3-o 

3-5 

4.0 

4-4 

49 

5-2 

10.5 

iS-7 

21.0 

•  5 

.6 

.  i 

.6 

3-1 

3-7 

4- 

1 

4-7 

5° 

5-6 

ii.  i 

16.7 

22.2 

.6 

.7 

,2 

.8 

3-3 

3-9 

4- 

4 

5.0 

Si 

5-9 

ii.  8 

17.7 

23-6 

.6 

.8 

•4 

•5 

3-5 

4-i 

4- 

7 

5-3 

52 

6.2 

12.5 

18.7 

25.0 

.6 

•9 

.  $ 

3-  I 

3-7 

4.4 

5- 

o 

5-6 

53 

6.6 

13.2 

19.8 

26.5 

•  7 

•3 

.0 

.6 

3-3 

4.0 

4-6 

5- 

3 

6.0 

54 

7.0 

14.0 

21.0 

28.1 

•  7 

•4 

.  i 

.8 

3-5 

4.2 

4-9 

5-6 

6-3 

55        7.4      14.9      22.3 

29-7 

.7 

•  5 

•2    3 

•o    3-7      4-5        5-2        5- 

9        6.7 

AZIMUTH 


TABLE    XIII.  — VALUES     OF 
FOR   COMPUTING   THE 
ALTITUDE. 

J/6  p"1  sin  i'  sin2  t  tan  h  for/  =  i°  10' 


A' 


TABLE    XIII    (A). 


Latitudes  . 

Equation  of  time. 

Hour 
angle 

30° 

40° 

50° 

April  15,  om 
May  15,  —3111.8 

May  i  ,  —  311.0 
June    ,  -2111.5 

June  15,  om 

July    ,  +3m.5 

0° 

15 
30 
45 
60 
75 
90 
105 

120 

.0 

.0 

.1 

.2 

•3 
•4 
•4 
•4 
•  3 

.0 
.0 
.2 

•  3 
•  5 
.6 
.6 
.6 
•4 

.0 

.  i 

.2 

•4 
•  7 
.8 
•9 
.8 
.6 

July  26  +  611.3 
Aug.  15,  +4m.4 
Sept.  15,  -4m.8 
Oct.  15,  -1411.1 
Nov.  15,  -15111.3 
Dec.  24,  cm 
Jan.  15,  +  Qm.2 
Feb.  12,  +1411.4 
Mar.  15,  +911.1 

Aug.     ,  +  6m.i 
Sept.  i,om 

Oct.       ,    -IQH.2 

Nov.  3,  -1611.3 
Dec.     ,  —  iom.9 
Jan.     ,  +311.2 
Feb.    ,  +13:11.6 
Mar.    ,  +1211.5 
April  i,  +411.0 

135 

.2 

.3 

•4 

150 

.  i 

.  i 

.  2  ' 

165 

.0 

.0 

.1 

180 

.0 

.0 

.O 

For  an  increase  of  i'  in  p  the  term  increases  about  3  %. 


TABLE    XIII    (B), 
Sines  of  Azimuth  and  Hour  Angle. 


0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

0° 

.086 

80° 

10 

.240 

.281 

•318 

•352 

•384 

•4i3 

.440 

.466 

.490 

•513 

•534 

70 

20 

•534 

•554 

•574 

•  592 

.609 

.626 

.642 

•  657 

.672 

.686 

•609 

60 

30 

.609 

.712 

•724 

•  736 

.748 

•759 

.769 

•779 

.789 

•799 

.808 

50 

40 

.808. 

.817 

.826 

•834 

.842 

.849 

•  857 

.864 

.871 

.878 

.884 

40 

50 

.884 

.891 

.897 

.902 

.908 

•913 

.919 

•924 

.928 

•933 

.938 

30 

60 

.938 

•942 

.946 

•950 

•954 

•957 

.961 

.964 

.967 

.970 

•973 

20 

70 

•973 

.976 

.978 

.981 

•983 

•985 

.987 

.989 

•990 

•992 

•993 

10 

80 

•993 

•995 

.996 

•997 

.998 

•998 

•999 

•999 

.000 

.000 

.000 

0 

10° 

9° 

8° 

7° 

6° 

5° 

4° 

3° 

2° 

1° 

0° 

Cosines  of  Declination  and  Altitude. 


TABLES 


53 


TABLE    XIV.  —  FOR  FINDING  THE  CORRECTION   TO   THE   SUN'S 
DECLINATION.      (In  minutes  and  tenths.) 


8^  . 

Difference  for  i  hour. 

Proportional  parts. 

<*-"  £   C 

IP 

10" 

3" 

4" 

5" 

6" 

7" 

8" 

9" 

2O 

30" 

40" 

So" 

o 

.0 

.1 

.1 

.2 

.2 

.0 

.0 

.0 
.0 

.0 

.  o 

.0 

.0 

.0 

.0 

.0 

1 

.1 

•3 

•4 

•  5 

.6 

.0 

.0 

.0 

.1 

.1 

.  i 

.  I 

.1 

I 

.2 

•  3 

•  5 

•  7 

.8 

.0 

.0 

_, 

.  i 

.  I 

.2 

i 

.2 

•3 
•  3 

•4 
•  5 
.6 

.6 
•  7 
•9 

.8 

.0 
.2 

I.O 

.0 
.0 

.0 

.0 
.1 
.  I 

.  i 

.2 

.1 

.2 
.2 

.2 
.2 
.2 

.  2 
.2 

•3 

2 

•  3 

•  7 

.0 

•3 

i-7 

.0 

.1 

.2 

.2 

.2 

•3 

•3 

J 

•4 

.8 

.  i 

•  5 

1.9 

.0 

.1 

.  2 

.  2 

•3 

•3 

•  3 

£ 

•4 

.8 

•3 

•  7 

2.  I 

.0 

.1 

.2 

•3 

•3 

•3 

•4 

I 

•  5 

•9 

•4 

.8 

2-3 

.1 

.1 

.2 

•  3 

•3 

•4 

•4 

3 

•  5 

I.O 

•  5 

.0 

2-5 

.  I 

•3 

•  3 

•4 

•4 

•  5 

•  5 

.1 

.6 

.  2 

2-7 

.  I 

•  3 

•3 

•4 

•4 

•  5 

•^ 

.  6 

.  2 

.8 

•  3 

2.9 

.1 

•3 

•4 

•4 

•5 

.  r 

2 

.6 

•3 

•9 

•5 

3-1 

.1 

•3 

•  3 

•4 

•4 

•5 

.6 

4 

.7 

•3 

.0 

•7 

3-3 

.  I 

•3 

•  3 

•4 

•  5 

•  5 

.6 

•  7 

•4 

.  i 

.8 

3-5 

.1 

•3 

•4 

•4 

-5 

.6 

.6 

£ 

.8 

•  5 

•3 

3-o 

3-8 

.2 

•3 

•4 

•  5 

•  5 

.6 

•  7 

i 

.8 

.6 

•4 

3-2 

4.0 

.  2 

•  3 

•4 

•  5 

.6 

.6 

•  7 

5" 

.8 

.  7 

3-  3 

4  2 

2 

3 

3 

c 

.6 

.8 

•  9 

.8 

^6 

3-5 

4-4 

.2 

•3 

•4 

•4 

•  J 

•  5 

.6 

•  7 

I 

•  9 

.8 

.8 

3-7 

4.6 

.  2 

•  3 

•4 

•  5 

.6 

.6 

•  7 

'.8 

ji 

I    O 

•9 

•9 

3.8 

4.8 

.  2 

.3 

•4 

6 

M 

8 

•  O 

6 

I.O 

.0 

3-Q 

4.0 

5-o 

.2 

•3 

•4 

•5 

.6 

•  7 

.8 

•  9 

i 

I.O 

.1 

4-2 

5-2 

.2 

•  3 

-4 

•  5 

.6 

•  7 

.8 

•9 

•j^ 

1.  1 

.  2 

3-3 

4-3 

5-4 

.2 

•  3 

•4 

•5 

•  7 

.8 

•9 

.0 

* 

I.I 

•  3 

3-4 

4-5 

5-6 

.2 

•3 

•5 

.6 

•  7 

.8 

•9 

.0 

7 

1.2 

•  3 

3-5 

4-7 

5-8 

.2 

•4 

•5 

.6 

•  7 

.8 

•9 

i 

I.  2 

•4 
•  5 
.6 

3-6 
3-8 
3-9 

4.8 
5-0 

5-2 

6.0 
6-3 
6-5 

.2 
•3 

•  3 

•4 
•4 
•  4 

•  5 
•  5 
•  5 

.6 
.6 

•  7 

'a 

•9 
•9 
•9 

.0 

.0 
.0 

8 

1-3 

•  7 

4.0 

5-3 

6.7 

•  3 

•  4 

•  5 

•  7 

.8 

•  9 

.1 

£ 

1.4 

.8 

4.1 

5-5 

6.9 

•3 

•4 

.6 

•7 

.8 

.0 

.  i 

£ 

1.4 

.8 

4-3 

5-7 

7-  I 

•  3 

•4 

.6 

•  7 

•9 

.0 

.  i 

.3 

* 

i-5 

•9 

4-4 

5-8 

7-3 

•3 

•4 

.6 

•  7 

•9 

.0 

.  2 

•  3 

9 

i-5 

3-o 

4-5 

6.0 

7-5 

•3 

•5 

.6 

.8 

•  9 

.  i 

.  2 

•  4 

J 

i.  5 

3-  z 

4.6 

6.2 

7-7 

•  3 

•5 

.6 

.8 

•9 

.1 

.2 

•4 

* 

1.6 

3-2 

4.8 

6.3 

7-9 

•  3 

•  5 

.6 

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.0 

.  i 

•  3 

.4 

* 

1.6 

3-3 

4.9 

6-5 

8.1 

•3 

•5 

.7 

.8 

.0 

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•3 

•5 

IO 

i-7 

3-3 

5-o 

6.7 

8-3 

•3 

•5 

•  7 

.8 

.0 

.2 

•  3 

-5 

» 

i.  7 

3-4 

6.8 

8-5 

•  3 

•  5 

•  7 

•9 

.0 

.2 

•4 

.5 

i 

1.8 

3-5 

5-3 

7.0 

8.8 

•4 

•5 

•  7 

•9 

.1 

.2 

•4 

.6 

* 

1.8 

3-6 

5-4 

7-2 

9.0 

•4 

•  5 

•  7 

•9 

.  I 

•3 

•4 

.6 

ii 

1.8 

3-7 

5-5 

7-3 

9-2 

•4 

.6 

•  7 

•9 

.  I 

•  3 

.5 

•  7 

4 

1.9 

3-8 

5-6 

7-5 

9-4 

•4 

.6 

.8 

•9 

.1 

•  3 

•  5 

•  7 

•£ 

1.9 

3-8 

5-8 

7-7 

9-6 

•4 

.6 

.8 

I.O 

.  2 

.3 

.5 

.7 

* 

2.0 

3-9 

5-9 

7-8 

9.8 

•4 

.6 

.8 

i  .0 

.2 

•4 

.6 

.8 

12 

2.O 

4.0 

6.0 

8.0 

IO.O 

.2 

•4 

.6 

.8 

i  .0 

1.2 

1-4 

1.6 

1.8 

54  AZIMUTH 


TABLES  OF  THE  SUN'S  DECLINATION. 

The  following  tables  contain  the  sun's  declination  for  the  instant  of 
Greenwich  Mean  Noon  for  each  day  in  the  year  1916-9  inclusive.  To 
find  the  declination  for  any  other  instant  of  time  than  Greenwich  Mean 
Noon  increase  or  decrease  the  tabular  declination  by  an  amount  equal 
to  the  "difference  for  one  hour"  multiplied  by  the  number  of  hours 
elapsed  since  Greenwich  Mean  Noon.  If  the  watch  used  is  regulated 
to  standard  time,  the  Greenwich  Mean  Time  is  found  at  once  by  adding 
5h  for  Eastern  Time,  6h  for  Central  Time,  etc.  If  the  watch  keeps 
local  time,  the  Greenwich  Time  is  obtained  by  adding  the  west  longi- 
tude of  the  meridian  for  which  the  watch  is  regulated,  expressed  in 
hours,  minutes  and  seconds.  The  work  of  multiplying  the  "  difference 
for  one  hour  "by  the  number  of  hours  since  Greenwich  Noon  may  be 
avoided  by  employing  Table  XIV  on  page  53.  The  hours  since  Green- 
wich Noon  will  be  found  in  the  left-hand  column,  for  intervals  of  a  quar- 
ter of  an  hour.  The  " difference  for  one  hour"  is  given  at  the  top  for 
intervals  of  ten  seconds.  The  proportional  parts  for  seconds  of  declina- 
tion at  the  right  are  employed  exactly  as  in  the  other  tables  similarly 
arranged.  The  correction  to  the  declination  is  in  minutes  and  tenths 
of  minutes. 


TABLES  OF  THE  SUN'S  DECLINATION  55 


The  tables  for  1916-9  may  be  used  to  find  the  sun's  declination  for 
the  years  1920-3  by  applying  the  following  simple  rule : 

To  find  the  declination  for  any  day  in  the  year  1920  compute  the 
declination  for  the  corresponding  day  in  the  year  1916  but  for  one  hour 
later  in  the  day  than  the  actual  Greenwich  Mean  Time. 

To  find  the  declination  for  any  day  in  the  year  1921  employ  the  table 
for  the  year  1917;  time,  one  hour  later  than  the  given  time. 

To  find  the  declination  for  any  day  in  the  year  1922  employ  the  table 
for  the  year  1918;  time,  one  hour  later  than  the  given  time. 

To  find  the  declination  for  any  day  in  the  year  1923  employ  the  table 
for  the  year  1919;  time,  one  hour  later  than  the  given  time. 

EXAMPLE  : 

Find  the  sun's  declination  at  5  P.M.,  Eastern  Time,  May  19,  1920.  From  the  table  on 
PP-  56-7  the  declination  for  May  19,  1916,  at  Greenwich  Mean  Noon  is  N.  19°  45' -5  and 
the  "  diff.  for  one  hour  "  is  32". 2.  According  to  the  rule  we  must  compute  the  declination 
for  6  P.M.  Eastern  Time. 

6h  P.M.  Eastern  Time  +  5h  =  "h  P-M.,  G.  M.  T. 

May  19,  1916,  at  G.  M.  N.  decl.  =  N.  19°  45'. 5 

32".2  X  nh  (Table  XIV)  = $.£ 

Declination  at  sh  P.M.  May  19,  1920  =  N.  19°  5i'-4 

For  a  method  of  calculating  the  declination  for  any  date  by  employing 
the  ephemeris  for  the  preceding  years  see  p.  69. 


22 


1 

Declination.  Ei^for 

"?  .,,0.0.0    ,0,0.0.0.0    ,0.0.0.,.,    »,.,,.„„    n  ,.«>«,»      0.0000 

;     +     +     ;     +' 

1                              | 

SSSS2        SSSSc?       £££££       £££>;?;?      £c7S£S 

^           : 

« 

Dedination.  Diftta 

^ 

V****.*«*  sjsi    *45M**ilfri 

j»3!j 

6S^  S 

% 

2 

! 

Declination.  DH-for 

Ov 

|s^|  : 

-M4rid3«do6M«o«o   Sl!«^5vd^o6dd    do  odd 

»m«  : 

o 

^                                            ^  : 

1 

Declination.  W£ta 

o 

toioioioio    ^Jioioirsio     ioioir>ioio    10*^101010    10*0101010 

+*  *  "  "   + 

cO.to.O       ..-.COO       ^tocOO       ..toco.       Mcog^ 

^^ 

* 

February. 

1 

1 
Q 

t^,  M   rf  t^-     . 

.t^.toO>M      Ot^MWQ      t^  O   cs   M  oO      co  r^OO  t^  rf     O   co*O  O  to 

•?<?2-»  :    : 

CO\O    OMTj-vOt^OvOl-l        WMCMMM        WO    O>00    t^     O    •*  «    O  OO 

0 

ooo  oo  t-  :    : 

in    • 

C/2 

January. 

Declination.  ™Lte 

^n^^SS?  55555^5^^  ^53S5  °^Sd  : 

+       +       +       +       + 

+            + 

OOtOtO.       COCOCNMO        tOTj-COWM        OtO^fcOfN        QtOTfN*-! 

00    COCO    <NO       O 
10  •*  <N    M    10       Tj- 

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64 


AZIMUTH 


PLATES 


66 


AZIMUTH 


COKSTELLATIOXS  ABOUT  THE  NOBTH  POLE 


MISCELLANEOUS   RULES 
AND   TABLES 


CORRECTING   THE   WATCH. 

To  find  the  approximate  time,  for  correcting  the  sun's  declination: 

1.  Add  the  log  sin  azimuth  (Table  XIII  B)  to  the  log  cos  altitude 
and  from  the  sum  subtract  the  log  cos  declination.     The  result  is  the 
log  sin  hour  angle.     Convert  this  angle  into  hours,  minutes,  and  seconds. 
If  it  is  forenoon,  subtract  this  hour  angle  from  i2h  to  obtain  thfe  solar 
time. 

2.  Convert  the  time  thus  found  into  Mean  Time  by  adding  or  sub- 
tracting the  Equation  of  Time  (Table  XIII  A). 

3.  Convert  this  Mean  Time  into  Standard  Time  by  taking  the  differ- 
ence in  longitude  between  the  place  and  the  standard  meridian,  express- 
ing it  as  hours,  minutes,  and  seconds,  and  adding  it  to  the  mean  time  if 
the  place  is  west  of  the  Standard  Meridian,  subtracting  if  it  is  east.    The 
difference  between  this  and  the  average  watch  reading  is  the  error  of  the 
watch.* 

This  problem  might  be  applied  as  follows.  If  the  surveyor  is  far 
from  a  place  where  the  time  can  be  obtained  he  may  work  up  his  azimuth 
observation  as  previously  explained  and  then  compute  the  time  by  this 
method.  If  it  is  discovered  that  the  watch  is  largely  in  error  the  azimuth 
should  be  recomputed,  using  the  corrected  declination  of  the  sun.  If  the 
time  is  thus  computed  with  each  azimuth  observation  the  error  of  the 
watch  may  be  known  approximately  at  all  times. 

The  only  doubtful  case  in  the  above  solution  is  when  the  sun  is  about 
6n  east  or  west  of  the  meridian.  The  sine  will  be  the  same  for  the  hour 
angle  or  its  supplement.  To  remove  this  ambiguity  compute  the  altitude 
of  the  sun  when  the  hour  angle  is  6h,  which  is  done  by  adding  the  log 
sin  latitude  to  the  log  sin  declination.  The  result  is  the  log  sin  altitude 
at  the  instant  when  the  sun  is  6h  from  the  meridian.  If  the  observed 
altitude  is  less  than  this  computed  altitude  the  hour  angle  is  greater 
than  90°  and  vice  versa. 

*  If  it  is  desired  to  compute  the  sun's  hour  angle  without  first  computing  the  azimuth 
this  may  be  done  by  means  of  equation  [6]. 


68 


CORRECTING  THE  WATCH  69 

EXAMPLE.  From  example  2,  p.  12,  we  have  Declination  =  +  21°  40',  Altitude  = 
57°  20',  Azimuth  =  S.  59°  55'  E.,  Eastern  Time  ph  49™,  A.M.  The  longitude  is  approxi- 
mately 71°  01'  W.  Date,  July  15. 

log  sin  azimuth  =    .938  Standard  Meridian     75° 

log  cos  altitude  =    .732  Local  Meridian     71°   or' 

i .  670  Difference  =         3°   59' 

log  cos  declination  =    .969  =        15™  56s, 

log  sin  hour  angle  =    .701 
hour  angle  =  30° .  2 

«=     2hOIm 

Time  =  9h  59'" 
Equation  =     +6™ 

Mean  Time  =  ioh  osm 
Longitude  correction  16™ 

Eastern  Time        9h49m  showing  that  the  watch  was  correct. 

In  example  3,  p.  12,  the  declination  is  +  16°  50'  .7  and  the  latitude  is  42°  29'  .2. 
log  sin  declination  =  .462 
log  sin  latitude        =  .830 

log  sin  altitude        =  .292 
Altitude  at  6h          =n°.3- 

Hence  the  hour  angle  of  the  sun  at  the  time  of  the  observation  was  less  than  6h,  since  the 
observed  altitude  is  greater  than  n°.3. 

To  obtain  the  sun's  declination  on  any  day  at  G.  M.  N.  from  an  Almanac  of  the  pre- 
vious year.  Take  out  the  declination  for  G.  M.  N.  of  the  same  date  and  compute  its 
value  for  a  time  6h  earlier.  This  may  be  done  conveniently  by  taking  the  "  diff.  for  one 
hour  "  in  seconds,  moving  the  decimal  point  one  place  to  the  left,  and  calling  the  result 
minutes.  This  correction  is  to  be  added  if  the  declination  is  numerically  decreasing; 
subtracted  if  increasing.  The  resulting  declination  is  generally  correct  within  a  small 
fraction  of  one  minute. 

The  declination  may  be  made  more  accurate  by  applying  a  further  correction  as 
follows:  —  Take  two  tenths  of  the  "  diff.  for  one  hour,"  convert  it  into  minutes,  and  apply 
this  correction  the  opposite  way  from  the  preceding  correction.  Notice  that  this  correc- 
tion is  one  thirtieth  of  the  first  correction. 

On  a  leap  year,  after  March  i,  one  day  must  be  added  to  the  given  date  before  entering 
the  almanac.  For  example,  March  2,  1912,  would  correspond  to  March  3,  1911,  when 
applying  this  rule. 

EXAMPLES,  i.  It  is  desired  to  know  the  declination  of  the  sun  at  G.  M.  N.  Jan.  3, 
1912,  only  a  1911  almanac  being  available.  For  Jan.  3,  1911  the  declination  is  S.  22°  54' 
34".4  =  S.  22°  54'. 57.  The  diff.  for  one  hour  =  i3".75-  The  first  correction  =  i'.375. 
The  declination  is  numerically  decreasing,  so  the  correction  is  added,  giving  S.  22°  55'.94- 
To  make  the  second  correction  we  subtract  0.2  X  13.75  •*•  60  =o'.os,  giving  S.  22°  55'. &g. 
2.  To  find  the  declination  for  June  10,  1912  from  the  almanac  for  1911.  June  10,  1912, 
corresponds  to  June  n,  1911.  The  declination  for  June  n,  1911  is  N.  23°oi'.92|  the 
difference  for  ih  is  n".36.  The  declination  for  June  10,  1912  is  therefore  N.  23°  oi'.92  — 
i'.i4  =  N.  23°  oo'-78.  Making  the  second  correction  reduces  it  to  N.  23°  oo'.82. 


7° 


AZIMUTH 


TABLE    XVII.  — CORRECTIONS  FOR    REDUCING    SLOPE 
MEASUREMENTS    TO    HORIZONTAL. 


1 

ft. 

ft. 

ft. 

F 

set. 

1 

100 

200 

300 

IO 

2O 

30 

40 

50 

60 

70 

80 

90 

o°  30' 

.00 

.01 

.01 

40' 

.  OI 

.01 

.02 

.01 

.01 

.01 

50 

.01 

.02 

•03 



.01 

.01 

.01 

.01 

.01 

i°oo' 

.02 

•03 

•05 

.01 

.01 

.01 

.01 

.01 

.01 

IOX 

.02 

•P4 

.06 

.  OI 

.  OI 

.01 

.  OI 

.01 

.02 

.02 

20' 

•03 

•05 

.08 

.OI 

.OI 

.01 

.01 

.02 

.02 

.02 

.02 

30' 

.03 

.07 

.  IO 

.OI 

.OI 

.01 

.02 

.02 

.02 

.03 

.03 

40' 

.04 

.08 

•13 

v 

.01 

.01 

.02 

.02 

•  03 

•03 

•03 

.04 

50 

•05 

.  IO 

•15 

.01 

.01 

.02 

.02 

•  03 

•03 

.04 

.04 

•  05 

2°00' 

.06 

.  12 

.18 

.01 

.01 

.02 

•  02 

•  03 

.04 

.04 

•05 

•  05 

10' 

.07 

.14 

.21 

.01 

.01 

.02 

•03 

.04 

.04 

•05 

.06 

.06 

2o' 

.08 

•17 

•25 

.01 

.02 

.02 

•  03 

.04 

•05 

.06 

•07 

.07 

30' 

.10 

.29 

.01 

.02 

•03 

.04 

•05 

.06 

.07 

.08 

.09 

40' 

.11 

.22 

•32 

.01 

.02 

•03 

.04 

•05 

.06 

.08 

.09 

.  IO 

50 

.12 

.24 

•37 

.01 

.02 

.04 

•05 

.06 

.07 

.09 

.10 

.11 

3°oo' 

.14 

•27 

.41 

.01 

•03 

.04 

•05 

.07 

.08 

.10 

.11 

.12 

10' 

•15 

•31 

.46 

.02 

•03 

•05 

.06 

.08 

.09 

.11 

.12 

.14 

20' 

•17 

•34 

.02 

•03 

•05 

.07 

.08 

.10 

.12 

•14 

•15 

30' 

.19 

•37 

•56 

.02 

.04 

.06 

•07 

.09 

.  II 

•  '3 

•15 

•17 

40; 

.20 

.41 

.61 

.02 

.04 

.06 

.08 

.10 

.12 

.14 

.16 

.18 

.22 

•45 

.67 

.02 

.04 

•07 

.09 

.11 

•13 

.16 

.18 

.20 

4°oo' 

.24 

•49 

•73 

.02 

•05 

•07 

.  IO 

.12 

•15 

•  17 

•19 

.22 

10' 

.26 

•53 

•79 

•03 

•05 

.08 

.11 

•13 

.16 

.19 

.21 

.24 

20' 

.29 

•57 

.86 

•03 

•  06 

.09 

.11 

.14 

•17 

.20 

•23 

.26 

30' 

•31 

.62 

•92 

•03 

.06 

.09 

.12 

•IS 

.18 

.22 

•25 

.28 

40' 

•33 

.66 

•99 

•03 

.07 

.10 

•13 

•17 

.20 

•23 

•27 

•30 

•36 

•7i 

1.07 

.04 

.07 

.  ii 

.14 

.18 

.21 

•25 

.28 

•32 

5°  oo' 

•38 

.76 

.14 

.04 

.08 

.  ii 

•IS 

.19 

•23 

.27 

•30 

•34 

10' 

.41 

.81 

.22 

.04 

.08 

.12 

.16 

.  2O 

•24 

.28 

•33 

•37 

20' 

•43 

.87 

•  30 

.04 

.09 

•13 

•17 

.22 

.26 

•30 

•35 

•39 

3°r 

.46 

.92 

.38 

•05 

.09 

.14 

.18 

•23 

.28 

•32 

•37 

.41 

•49 

.08 

•47 

•05 

.10 

•  IS 

.20 

•24 

.29 

•34 

•39 

•44 

50' 

•52 

1.04 

•55 

•05 

.  IO 

.16 

.21 

.26 

•31 

•36 

.41 

•47 

6°oo' 

•55 

.10 

.64 

•05 

.  II 

.16 

.22 

.27 

•33 

•38 

•44 

•49 

10' 

•58 

.16 

•74 

•05 

.12 

•  17 

•23 

.29 

•35 

.41 

.46 

•52 

20' 

.61 

.22 

•83 

.06 

.12 

.18 

•24 

•31 

•37 

•43 

•49 

•55 

30' 

.64 

.29 

•93 

.06 

•13 

.19 

.26 

•32 

•39 

•45 

•58 

40' 

.68 

•35 

•03 

.07 

.14 

.20 

•27 

•34 

.41 

•47 

•54 

.61 

50' 

.42 

•13 

.07 

.14 

.  21 

.28 

•36 

•43 

•50 

•57 

.64 

J 

EXAMPLE.  —  Required  the  horizontal  distance  for  a  slope  distance  of  272.46  ft., 
vertical  angle  of  4°  16'.  For  4°  10'  the  correction  is  0.53  +  0.19  =  072  ft.  Inter- 
polating between  4°  10'  and  4°  20'  for  distance  300  we  find  0.04  as  the  increase  for  6'. 
Hence  correction  =  0.76,  and  horizontal  distance  =  271.70  ft. 


MISCELLANEOUS  TABLES 


TABLE   XVII — (Continued). 


M 

I 

ft. 

ft. 

ft. 

Feet. 

i 

100 

200 

300 

IO 

2O 

30 

40 

So; 

60 

70 

80 

QO 

7°oo' 

•75 

1.49 

2.24 

.07 

•IS 

.22 

•30 

•37 

•45 

•52 

.60 

.67 

05' 

.76 

•53 

2.29 

.08 

•15 

•23 

•31 

•38 

.46 

•  53 

.61 

.69 

10' 

•78 

•56 

2.34 

.08 

.16 

•23 

•31 

•39 

•47 

•55 

.62 

•70 

15' 

.80 

.60 

2.40 

.08 

,16 

.24 

•32 

.40 

.48 

•56 

.64 

.72 

20' 

.82 

.64 

2-45 

.08 

.16 

•25 

•33 

.41 

•49 

•57 

•65 

•74 

25' 

.84 

.67 

2.51 

.08 

•17 

•25 

•33 

.42 

•50 

•59 

•67 

•75 

30' 

.86 

•71 

2-57 

.09 

•  17 

.26 

•34 

•43 

•Si 

.60 

.68 

•77 

35' 

.88 

•75 

2.63 

.09 

.18 

.26 

•35 

•44 

•53 

.61 

•70 

•79 

4o' 

.89 

•79 

2.68 

.09 

.18 

•27 

•36 

•45 

•54 

•63 

•72 

.80 

45' 

.91 

1.83 

2.74 

.09 

.18 

•27 

•37 

.46 

•55 

.64 

•73 

.82 

So' 

•93 

1.87 

2.80 

.09 

.19 

.28 

•37 

•47 

•  56 

.65 

•75 

.84 

55' 

•95 

1.91 

2.86 

.10 

.19 

.29 

•38 

.48 

•57 

.67 

.76 

.86 

8°oo' 

•97 

i-95 

2.92 

.10 

.19 

•29 

•39 

•49 

•58 

.68 

•78 

.88 

05' 

•99 

1.99 

2.98 

.10 

.20 

•30 

.40 

•50 

.60 

.70 

.80 

.89 

10' 

I.OI 

2.03 

3-04 

.10 

.20 

•30 

.41 

•51 

.61 

.71 

.81 

.91 

15' 

1.04 

.07 

.10 

.21 

.41 

•52 

.62 

•  72 

•83 

•93 

20' 

i.  06 

.11 

3-17 

.11 

.21 

•32 

.42 

•53 

•63 

•74 

•84 

•95 

25' 

i.  08 

•15 

3-23 

.11 

.22 

•32 

•43 

•34 

•65 

•75 

.86 

•97 

3o' 

.10 

.20 

3-29 

.11 

.22 

•33 

•44 

•55 

.66 

•77 

.88 

•99 

35' 

.12 

.24 

3.36 

.11 

.22 

•34 

•45 

•56 

.67 

•78 

.90 

I.OI 

40' 

.14 

.28 

3-43 

.11 

•23 

•34 

.46 

•57 

.69 

.80 

.91 

1.03 

45/ 

.16 

2-33 

3-49 

.12 

•23 

•47 

•58 

.70 

.81 

•93 

1.05 

.19 

2-37 

3-56 

.12 

.24 

.36 

•47 

•59 

•83 

•95 

1.07 

55' 

.21 

2.42 

3-63 

.12 

.24 

•36 

.48 

.60 

•73 

•85 

•97 

1.09 

9°  oo' 

•23 

2.46 

3.69 

.12 

•25 

•37 

•49 

.62 

•74 

.86 

.98 

i.  ii 

05' 

•25 

2.51 

3-76 

•13 

•25 

•38 

•50 

•63 

•75 

.88 

1.  00 

1.13 

10' 

.28 

2-55 

3-83 

•13 

.26 

•38 

•51 

.64 

•77 

.89 

1.02 

i.iS 

15^ 

•30 

2.60 

3-90 

•13 

.26 

•39 

•52 

•65 

.78 

.91 

1.04 

1.17 

•32 

2-65 

3-97 

•13 

.26 

.40 

•53 

.66 

•79 

•93 

1.  06 

1.19 

25' 

•35 

2.70 

4.04 

•13 

.27 

.40 

•54 

•67 

.81 

•94 

1.08 

1.  21 

30' 

•37 

2.74 

4.11 

.14 

•27 

.41 

•55 

.69 

.82 

.96 

.10 

1.23 

35' 

.40 

2.79 

4.19 

.14 

.28 

.42 

•56 

.70 

.84 

.98 

.12 

1.26 

40' 

.42 

2.84 

4.26 

.14 

.28 

•43 

•57 

•71 

•85 

•99 

.14 

1.28 

45' 

•44 

2.89 

4-33 

.14 

•29 

•43 

•58 

.72 

•87 

I.OI 

.16 

1.30 

50' 

•47 

2.94 

4.41 

•15 

.29 

•44 

•59 

•73 

.88 

1.03 

.18 

1-32 

55' 

•49 

2-99 

4.48 

•15 

•30 

•45 

.60 

•75 

•90 

1.05 

.20 

1-34 

10°  oo' 

1-52 

3-04 

4.56 

•15 

•30 

.46 

.61 

.76 

.91 

1.06 

1.22 

1-37 

To  reduce  slope  measurements  to  horizontal  when  the  angle  is  greater  than  10°,  add  the 
log  vers.  of  the  angle  (Table  VI)  to  the  log  slope  distance  (Table  V).  The  sum  is  the  log 
correction.  Subtract  the  correction  from  the  slope  distance . 

To  reduce  a  slope  measurement  to  horizontal  when  the  difference  in  elevation  of  the 
ends  of  the  tape  is  given:  Divide  the  square  of  the  height  by  twice  the  slope  length  and 
subtract  the  result  from  the  slope  length. 

EXAMPLE.  Height,  8.1  feet,  slope  length,  85  feet.  (8.1)2^-170  =  0.38.  Horizontal 
distance  =  84.62  feet. 

If  the  height  is  large  compared  with  the  distance  make  a  second  computation,  dividing 
the  square  of  the  height  by  the  sum  of  the  slope  length  and  the  approximate  horizontal 
distance.  Use  this  correction  instead  of  the  first. 

EXAMPLE.  Height,  20  feet,  slope  length,  loofeet.  (2o)2  -;-  200  =  2.00  feet.  Approxi- 
mate horizontal  distance  =  98.0  feet.  (2o)2  -H  198.0  =  2.02.  Horizontal  distance  =  97.98 
feet. 


72  AZIMUTH 

TABLE  XVIII.  —  INCHES  IN     DECIMALS  OF  A  FOOT. 


In. 

0 

X 

3 

3 

4 

5 

6 

7 

8 

9 

10 

ii 

iln. 

0 

Feet 

.0833 

.I667 

.2500 

•3333 

.4167 

.5000 

.5833 

.6667 

.7500 

•  8333 

.9167 

0 

A 

.0026 

.0859 

•1693 

.2526 

•  3359 

•4193 

.5026 

•5859 

•  6693 

.7526 

•  8359 

•9193 

A 

A 

.0052 

.0885 

.1719 

•  2552 

.3385 

.4219 

•5052 

•5885 

.6719 

•7552 

•8385 

.9219 

A 

A 

.0078 

.0911 

•1745 

•  2578 

•34" 

•  4245 

.5078 

•59" 

•6745 

•7578 

.8411 

•9245  & 

1 

.0104 

.0938 

.1771 

.2604 

•3438 

.4271 

.5104 

•5938 

.6771 

.7604 

•  8438 

•9271  * 

& 

.0130 

.0964 

.1797 

.2630 

•3464 

.4297 

•5130 

•5964 

.6797 

.7630 

.8464 

•9297   32 

A 

.0156 

.0990 

.1823 

.2656 

•  349° 

•  4323 

•5156 

•5990 

.6823 

.7656 

.8490 

•9323  i3s 

3^ 

.0182 

.1016 

.1849 

.2682 

•  35i6 

•4349 

.5182 

.6016 

.6849 

.7682 

-8516 

•  9349  35 

i 

.0208 

.1042 

•1875 

.2708 

•  3542 

•  4375 

.5208 

.6042 

.6875 

.7708 

8542 

•  9375  £ 

A 

.0234 

.1068 

.  1901 

•  2734 

3568 

.4401 

.5=34 

.6068 

.6901 

7734 

8568 

.9401  & 

A 

.0260 

.1094 

.1927 

.2760 

3594 

•  4427 

.5260 

.6094 

.6927 

7760 

8594 

9427  i53 

U 

.0286 

.1120 

•1953 

.2786 

3620 

•  4453 

.5286 

.6120 

•6953 

7786 

8620 

9453  & 

i 

•  0313 

.1146 

.1979 

.2813 

3646 

•4479 

•5313 

.6146 

.6979 

7813 

8646 

9479  1 

if 

•0339 

.1172 

.2005 

.2839 

3672 

45<>5 

•5339 

.6172 

.7005 

7839 

8672 

9505  i3 

A 

•  0365 

.1198 

.2031 

.2865 

3698 

4531 

•  5365 

.6198 

•  7031 

7865 

8698 

9531   /B 

?,:} 

.0391 

.1224 

.2057 

.2891 

3724 

•4557 

5391 

.6224 

•7Q57 

7891 

8724 

9557  M 

i 

.0417 

.1250 

.2083 

.2917 

3750 

4583 

5417 

.6250 

.7083 

7917 

8750- 

9583 

i 

U 

•  0443 

.1276 

.2109 

•  2943 

3776 

4609 

5443 

.6276 

.7109 

7943 

8776 

9609 

a 

T9* 

.0469 

.I3O2 

•2135 

.2969 

3802 

4635 

5469 

.6302 

7135 

7969 

8802 

9635 

T°s 

U 

•  °495 

.1328  .2l6l  .2995 

3828 

4661 

5495 

.6328 

7161 

7995 

8828 

9661 

*§ 

i 

.0521 

•1354 

.2188 

.3021 

3854 

4688 

5521 

•  6354 

.7188 

8021 

8854 

9688 

i 

3^ 

.0547 

.1380 

.2214 

•3°47 

3880 

4714 

5547 

.6380 

.7214 

8047 

8880 

97H 

& 

fi 

.0573 

.1406 

.2240  .3073 

3906 

4740 

5573 

.6406 

7240 

8073 

8906 

9740 

tt 

H 

•  0599 

.1432 

.2266 

•3099 

3932 

4766 

5599 

.6432 

.7266 

8099 

8932 

9766 

ff 

i 

.0625 

.1458 

.2292 

•3125 

3958 

4792 

5625 

.6458 

7292 

8125 

8958 

9792 

i 

H 

.0651 

.1484 

.2318 

•3151 

3984 

4818 

5651 

.6484 

73i8 

8151 

8984 

9818 

§5 

13 

.0677 

.1510 

•2344 

.3177 

4010 

4844 

5677 

.6510 

7344 

8i77 

9010 

9844 

ii 

H 

.0703 

•1536 

.2370 

.3203 

4036 

4870 

5703 

.6536 

7370 

8203 

9036 

9870 

fi 

i 

.0729 

.1563 

.2396 

.3229 

4063 

4896 

5729 

.6563 

739s 

8229 

9063 

9896 

1 

§5 

•°755 

.1589 

.2422 

•  3255 

.4089 

4922 

5755 

.6589 

7422 

8255 

9089 

9922 

§§ 

H 

.0781 

.1615 

.2448 

.3281 

4"5 

4948 

.578i 

.6615 

7448 

8281 

9«5 

9948 

15 

H 

.0807 

.  1641 

.2474 

•3307 

4141 

4974 

•5807 

.6641 

7474 

8307 

9141 

9974 

& 

In. 

0 

I 

a 

3 

4 

5 

6 

7 

8 

9 

10 

ii  'In. 

MISCELLANEOUS  TABLES^  :  • 


73 


TABLE  XIX.— CONSTANTS. 


Number. 

Logarithm. 

Ratio  of  circumference  to  diameter  
Base  of  hyperbolic  logarithms 

3-  HIS9 
2   71828 

0.49715 
o  4.3420 

Modulus  of  common  system  of  logs 

o.  43420 

o.  63778—10 

Length  of  seconds  pendulum  at  N.  Y. 
(inches)  

30.  1017 

I.  CTQ22O 

Acceleration  due  to  gravity  at  N.  Y  
Cubic  inches  in  i  U.  S.  gallon  

32-i5949 

231. 

1-50731 

2.  36361 

Cubic  feet  in  i  U.  S.  gallon  

o.  1337 

9-  12613—10 

U.  S.  gallons  in  i  cubic  foot 

7   480? 

o  87303 

Pounds  of  water  in  i  cubic  foot 

62  t; 

i.  7QC88 

Pounds  of  water  in  i  U.  S.  gallon   .    ... 

8.  3SS 

o.  02102; 

Pounds  per  square  inch  due  to  i  atmos..  .  . 
Pounds  per  square  inch  due  to  i  foot 
head  of  water  
Feet  of  head  for  pressure  of  i  pound  per 
square  inch  

14.7 
0-434 

2    3O4 

1.16732 

9-  63749-io 
o  36248 

Inches  in  i  centimeter 

O    3Q37 

9CQCI  7—IO 

Centimeters  in  i  inch 

2     ^4OO 

o  40483 

Feet  in  i  meter        

3    2808 

o  sisoS 

Meters  in  i  foot  i  

o  304.8 

9  48402—10 

Miles  in  r  kilometer  :  .    .    . 

o  62137 

o  7033^—10 

Kilometers  in  i  mile  
Square  inches  in  i  square  centimeter..  .  . 
Square  centimeters  in  i  square  inch.  .  .  . 
Square  feet  in  i  square  meter  
Square  meters  in  i  square  foot  
Cubic  feet  in  i  cubic  meter  

1.60935 
0.155° 

6.4520 
10.  764 
0.09290 
3C.  31^6 

o.  20665 
9.  19033-10 
o.  80969 

1.03197 
8.96802-10 

I.  ^4797 

Pounds  (av.)  in  i  kilogram  

2.  2046 

o.  34333 

Kilograms  in  i  pound  (av.)  
Ft.-lbs.  in  i  kilogram-meter             

0-4536 

7    2  3  308 

9.  65667-10 
o.  8^032 

Natural  sine  of  o°  oo'  01"  =  .00000485;  log 
Natural  sine  of  o°  01'  =  .00029089;  log 

Natural  sine  of  o°  i' 
Natural  sine  of  o°  oo'  01" 


4-68557 

6.46373 
o.  03  ft. 

100  ft. 

0.3  inch 
i  mile 


/rf) 


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